https://peda.net/id/94295a3406c 2019-11-14T11:31:47+02:00 https://peda.net/:static/532/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/30d0baee1be 2019-12-11T09:38:14+02:00 <span>433</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%7B%2C%7Dy%5Cright)%3D%5Cfrac%7B5%7D%7Bx%5E2%2By%5E2%2B1%7D" alt="f\left(x{,}y\right)=\frac{5}{x^2+y^2+1}"/></div> <div>a) Ratkaistaan yhtälö</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%7B%2C%7Dy%5Cright)%3D1" alt="f\left(x{,}y\right)=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B5%7D%7Bx%5E2%2By%5E2%2B1%7D%3D1" alt="\frac{5}{x^2+y^2+1}=1"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=5%3Dx%5E2%2By%5E2%2B1" alt="5=x^2+y^2+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2By%5E2%3D4" alt="x^2+y^2=4"/></div> <div>Funktio f saa arvon 1 ympyrän <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2By%5E2%3D4" alt="x^2+y^2=4"/>pisteissä.<br/> <div>b) Funktio f(x,y) arvo on suurin, kun positiivisia arvoja saava nimitäjä saa pienimmän arvonsa<img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%5Cge0%5C%20ja%5C%20y%5E2%5Cge0" alt="x^2\ge0\ ja\ y^2\ge0"/>aine, joten <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2By%5E2%2B1%5Cge1" alt="x^2+y^2+1\ge1"/>.</div> <div>Nimittäjän pienin arvo on siis 1</div> <div>Funktion suurin arvo on 5.</div> </div> 2019-12-11T09:38:14+02:00 https://peda.net/id/eb01e49c174 2019-12-05T11:45:06+02:00 <span>4.1 Käänteisfunktio ja sen derivaatta</span> <div>Määritelmä</div> <div>Olkoon funktiot f:A(Määrittely joukko)→B(arvojoukko) ja g:B→A</div> <div>f ja g ovat toistensa käänteisfunktiota, jos </div> <div>g(f(x))=X kaikilla x∈A </div> <div>f(g(y))=Y kaikilla y∈B</div> <div> </div> <div>402 </div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D3x%7B%2C%7D%5C%20g%5Cleft(x%5Cright)%3D%5Cfrac%7Bx%7D%7B3%7D" alt="f\left(x\right)=3x{,}\ g\left(x\right)=\frac{x}{3}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(f%5Cleft(x%5Cright)%5Cright)%3Dg%5Cleft(3x%5Cright)%3D%5Cfrac%7B3x%7D%7B3%7D%3Dx" alt="g\left(f\left(x\right)\right)=g\left(3x\right)=\frac{3x}{3}=x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(g%5Cleft(x%5Cright)%5Cright)%3D3%5Cleft(%5Cfrac%7Bx%7D%7B3%7D%5Cright)%3Dx" alt="f\left(g\left(x\right)\right)=3\left(\frac{x}{3}\right)=x"/><br/> <div>Ovat toistensa käänteisfunktiota</div> <div> </div> <div>- <img src="https://math-demo.abitti.fi/math.svg?latex=f%5E%7B-1%7D%5Cleft(y%5Cright)%3Dx%5C%20joss%5C%20f%5Cleft(x%5Cright)%3Dy" alt="f^{-1}\left(y\right)=x\ joss\ f\left(x\right)=y"/></div> <span>Esim. </span><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3D7" alt="f\left(1\right)=7"/><span>,</span><img src="https://math-demo.abitti.fi/math.svg?latex=f%5E%7B-1%7D%5Cleft(7%5Cright)%3D1" alt="f^{-1}\left(7\right)=1"/> <div>- Jos funktio määrittelyjoukko on jokin väli ja funtki oon monotoninen tälä välillä, on funtkiolal käänteisfunktio</div> <div>- Funktion f arojoukko on funktion<img src="https://math-demo.abitti.fi/math.svg?latex=f%5E%7B-1%7D" alt="f^{-1}"/>määrittelyjoukko ja f:n määrittelyjoukko on funktion<img src="https://math-demo.abitti.fi/math.svg?latex=f%5E%7B-1%7D" alt="f^{-1}"/>arvojoukko</div> <div>- Käänteisfunktion lausekkeen muodostaminen:</div> <div>1. Merkitse f(x)=y ja ratkaise siitä x</div> <div>2. Vaihda x⇔y</div> <div>3. Kirjoita muodossa <img src="https://math-demo.abitti.fi/math.svg?latex=f%5E%7B-1%7D%5Cleft(x%5Cright)%3Dy" alt="f^{-1}\left(x\right)=y"/></div> <div>Esimerkki. f(x)=x²+x f:n[0,∞[→[3,∞[</div> <div>a) Osoita, että funktiolla on käänteisfunktio. Mikä on sen määrittelyjoukko?</div> <div>b) Määritä <img src="https://math-demo.abitti.fi/math.svg?latex=f%5E%7B-1%7D%5Cleft(4%5Cright)" alt="f^{-1}\left(4\right)"/></div> <div>c) Määritä känteisfunktio laseke</div> <div>Ratkaisu:</div> <div>a) </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D2x" alt="f'\left(x\right)=2x"/><br/> <div>Koska x≥0, niin f'(x)≥0 ja f(x) on kasvava eli monotoninen välillä [0,∞[. Siten funktiolla on olemassa käänteisfunktio, jonka määrittelyjoukko on [3,∞[</div> 2019-12-05T11:45:06+02:00 https://peda.net/id/1ffe18d0173 2019-12-05T10:49:19+02:00 <div>3.2</div> <div>- Jos statunnaismuuttuja X voi saada mink tahansa arvon joltakin lukusuoran väliltä, se on jatkuva satunnaismuuttuja</div> <div>- Funktio ∈ℝ→ℝ on satunnaismuuttujan X tiheysfunktio joss f(x) ≥ 0 JA<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7Df%5Cleft(x%5Cright)dx%3D1" alt="\int_{-\infty}^{\infty}f\left(x\right)dx=1"/></div> <div>- Funktio F:ℝ→ℝ, F(x)=P(X≤x) on satunnaismuuttuja X kertymäfuntio. Se kertoo, millä todennäköiisyydellä satunnaismuuttujan arvo on piene,pi tai yhtä suuri kuin x. Koska todennäköisyydet saadaan tiheysfunktion alle jäävästä pinta-alasta, niin<span> </span><img src="https://math-demo.abitti.fi/math.svg?latex=F(x)%3DP(X%5Cle%20x)%3D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7Df%5Cleft(t%5Cright)dt" alt="F(x)=P(X\le x)=\int_{-\infty}^{\infty}f\left(t\right)dt"/>(tiheysfunktio)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P(X%5Cge%20a)%3D1-P(x%5Cle%20a)%3D1-F(a)" alt="P(X\ge a)=1-P(x\le a)=1-F(a)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(a%5Cle%20X%5Cle%20b%5Cright)%3DF%5Cleft(b%5Cright)-F%5Cleft(a%5Cright)" alt="P\left(a\le X\le b\right)=F\left(b\right)-F\left(a\right)"/></div> 2019-12-05T10:49:19+02:00 https://peda.net/id/6771873e166 2019-12-04T09:05:07+02:00 <div>Epäoleellinen intergraali on määrätty integraali, jonka alarajana on -∞ ha/tai ylärajana +∞. Ala tai ykärajana voi myös olla luku, jossa integroitava funktio ei ole määritelty.</div> <div>Esimerkiksi seuraavat ovat epäolellisisa integraaleja.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5E2%5Cleft(2x%5E2-x%5Cright)dx" alt="\int_{-\infty}^2\left(2x^2-x\right)dx"/><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-3%7D%5E%7B%5Cinfty%7D%5Cleft(4x%5E3%2B8x%5E2%5Cright)dx" alt="\int_{-3}^{\infty}\left(4x^3+8x^2\right)dx"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%5Cfrac%7Bx%7D%7Bx%5E2%2B1%7Ddx" alt="\int_{-\infty}^{\infty}\frac{x}{x^2+1}dx"/><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-1%7D%5E0%5Cfrac%7B1%7D%7Bx%7Ddx%7B%2C%7D%5C%20ei%5C%20ole%5C%20m%C3%A4%C3%A4ritelty%7B%2C%7D%5C%20kun%5C%20x%3D0" alt="\int_{-1}^0\frac{1}{x}dx{,}\ ei\ ole\ määritelty{,}\ kun\ x=0"/></div> <div>Esimerkki. Laske</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5E2%5Cleft(2x%5E2-x%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-%5Cinfty%7D%7D%5E2%5Cleft(2%5Ccdot%5Cfrac%7B1%7D%7B2%2B1%7Dx%5E%7B2%2B1%7D-%5Cfrac%7B1%7D%7B1%2B1%7Dx%5E%7B1%2B1%7D%5Cright)%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-%5Cinfty%7D%7D%5E2%5Cleft(%5Cfrac%7B2%7D%7B3%7Dx%5E2-%5Cfrac%7B1%7D%7B2%7Dx%5E2%5Cright)" alt="\int_{-\infty}^2\left(2x^2-x\right)dx=\bigg/_{\!\!\!\!\!{-\infty}}^2\left(2\cdot\frac{1}{2+1}x^{2+1}-\frac{1}{1+1}x^{1+1}\right)=\bigg/_{\!\!\!\!\!{-\infty}}^2\left(\frac{2}{3}x^2-\frac{1}{2}x^2\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cfrac%7B2%7D%7B3%7D%5Ccdot2%5E3-%5Cfrac%7B1%7D%7B2%7D%5Ccdot2%5E2%5Cright)-%3F%3F%3F" alt="=\left(\frac{2}{3}\cdot2^3-\frac{1}{2}\cdot2^2\right)-???"/></div> <span>Ongelma kierretään raja-arvon avulla.</span> <div>Integroinnissa on käytetty kaavaa: <img src="https://math-demo.abitti.fi/math.svg?latex=Kun%5C%20n%5Cne-1%7B%2C%7D%5C%20%5Cint_%7B%20%7D%5E%7B%20%7Dx%5Endx%3D%5Cfrac%7B1%7D%7Bn%2B1%7Dx%5E%7Bn%2B1%7D%2BC" alt="Kun\ n\ne-1{,}\ \int_{ }^{ }x^ndx=\frac{1}{n+1}x^{n+1}+C"/></div> <div>Määritelmä</div> <div>Välillä [a,∞[ jatkuvan funktion f epäolellinen integraali on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_a%5E%7B%5Cinfty%7Df%5Cleft(x%5Cright)dx%3D%5Clim_%7Bt%5Crightarrow%5Cinfty%7D%5Cleft(%5Cint_a%5Etf%5Cleft(x%5Cright)dx%5Cright)" alt="\int_a^{\infty}f\left(x\right)dx=\lim_{t\rightarrow\infty}\left(\int_a^tf\left(x\right)dx\right)"/>, jos raja-arvo on olemassa.</div> <div>Tällöin epäolellinen integraali suppenee. Jos äärellistä raja-arvoa ei ole, epäoleellinen integraali hajaantuu. </div> <div>Vastaavasti välillä ]-∞, b] jatkuvan funktion f epäoleellinen integraali on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5Ebf%5Cleft(x%5Cright)dx%3D%5Clim_%7Bt%5Crightarrow-%5Cinfty%7D%5Cleft(%5Cint_t%5Ebf%5Cleft(x%5Cright)dx%5Cright)" alt="\int_{-\infty}^bf\left(x\right)dx=\lim_{t\rightarrow-\infty}\left(\int_t^bf\left(x\right)dx\right)"/></div> <div>Esimerkki. Laske</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5E2%5Cleft(2x%5E2-x%5Cright)dx%3D%5Clim_%7Bt%5Crightarrow-%5Cinfty%7D%5Cleft(%5Cint_t%5E2%5Cleft(2x%5E2-x%5Cright)dx%5Cright)" alt="\int_{-\infty}^2\left(2x^2-x\right)dx=\lim_{t\rightarrow-\infty}\left(\int_t^2\left(2x^2-x\right)dx\right)"/></div> <div>Lasketaan ensin interaali<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_t%5E2%5Cleft(2x%5E2-x%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!t%7D%5E2%5Cfrac%7B2%7D%7B3%7Dx%5E3-%5Cfrac%7B1%7D%7B2%7Dx%5E2%3D%5Cleft(%5Cfrac%7B16%7D%7B3%7D-2%5Cright)-%5Cleft(%5Cfrac%7B2%7D%7B3%7Dt%5E2-%5Cfrac%7B1%7D%7B2%7Dt%5E2%5Cright)%3D%5Cfrac%7B10%7D%7B3%7D-%5Cfrac%7B2%7D%7B3%7Dt%5E3%2B%5Cfrac%7B1%7D%7B2%7Dt%5E2" alt="\int_t^2\left(2x^2-x\right)dx=\bigg/_{\!\!\!\!\!t}^2\frac{2}{3}x^3-\frac{1}{2}x^2=\left(\frac{16}{3}-2\right)-\left(\frac{2}{3}t^2-\frac{1}{2}t^2\right)=\frac{10}{3}-\frac{2}{3}t^3+\frac{1}{2}t^2"/></div> <div>Lasketaan seuraavasksi raja-arvo</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bt%5Crightarrow-%5Cinfty%7D%5Cleft(%5Cint_t%5E2%5Cleft(2x%5E2-x%5Cright)dx%5Cright)%3D%5Clim_%7Bt%5Crightarrow-%5Cinfty%7D%5Cleft(-%5Cfrac%7B2%7D%7B3%7Dt%5E3%2B%5Cfrac%7B1%7D%7B2%7Dt%5E2%2B%5Cfrac%7B10%7D%7B3%7D%5Cright)" alt="\lim_{t\rightarrow-\infty}\left(\int_t^2\left(2x^2-x\right)dx\right)=\lim_{t\rightarrow-\infty}\left(-\frac{2}{3}t^3+\frac{1}{2}t^2+\frac{10}{3}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bt%5Crightarrow%5Cinfty%7D%5Cleft(t%5E3%5Cleft(-%5Cfrac%7B2%7D%7B3%7D%2B%5Cfrac%7B1%7D%7B2t%7D%2B%5Cfrac%7B10%7D%7B3t%5E3%7D%5Cright)%5Cright)%3D%5Cinfty" alt="\lim_{t\rightarrow\infty}\left(t^3\left(-\frac{2}{3}+\frac{1}{2t}+\frac{10}{3t^3}\right)\right)=\infty"/></div> <div>Epäoleellinen integraali <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5E2%5Cleft(2x%5E2-x%5Cright)dx" alt="\int_{-\infty}^2\left(2x^2-x\right)dx"/>hajaantuu.</div> 2019-12-04T09:05:07+02:00 https://peda.net/id/9ade6aee10e 2019-11-27T09:14:52+02:00 Lukujonoa <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(a_k%5Cright)%3D%5Cleft(a_1%7B%2C%7D%5C%20a_2%7B%2C%7D%5C%20a_3%7B%2C%7D%5C%20...%5Cright)" alt="\left(a_k\right)=\left(a_1{,}\ a_2{,}\ a_3{,}\ ...\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7Da_k%3Da_1%2Ba_2%2Ba_3%2B..." alt="\sum_{k=1}^{\infty}a_k=a_1+a_2+a_3+..."/>kutsutaan sarjaksi. <div>Sarjan <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7Da_k%3Da_1%2Ba_2%2Ba_3%2B...%2Ba_n%2Ba_n" alt="\sum_{k=1}^{\infty}a_k=a_1+a_2+a_3+...+a_n+a_n"/>osasummia ovat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_1%3Da_1" alt="S_1=a_1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=S_2%3Da_1%2Ba_2" alt="S_2=a_1+a_2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=S_3%3Da_1%2Ba_2%2Ba_3" alt="S_3=a_1+a_2+a_3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3A" alt=":"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=S_n%3Da_1%2Ba_2%2Ba_3%2B...%2Ba_n" alt="S_n=a_1+a_2+a_3+...+a_n"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3A" alt=":"/></div> <div>Sarjan summa tarkoittaa raja-arvoa <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7D%5Cleft(a_1%2Ba_2%2B...%2Ba_n%5Cright)%3D%5Clim_%7Bn%5Crightarrow%5Cinfty%7DS_n" alt="\lim_{n\rightarrow\infty}\left(a_1+a_2+...+a_n\right)=\lim_{n\rightarrow\infty}S_n"/>eli osasummien jonon <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(S_n%5Cright)" alt="\left(S_n\right)"/>raja-arvoa.</div> <div>Sarja suppenee, jos (äärellinen) raja-arvo on olemassa. Muutoin sarja hajaantuu.</div> <div> </div> <div>270</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_n%3D%5Cfrac%7Bn%7D%7B2n%2B1%7D" alt="S_n=\frac{n}{2n+1}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7DS_n%3D%5Clim_%7Bn%5Crightarrow%5Cinfty%7D%5Cfrac%7Bn%7D%7B2n%2B1%7D%3D%5Clim_%7Bn%5Crightarrow%5Cinfty%7D%5Cfrac%7Bn%7D%7Bn%5Cleft(2%2B%5Cfrac%7B1%7D%7Bn%7D%5Cright)%7D%3D%5Clim_%7Bn%5Crightarrow%5Cinfty%7D%3D%5Cfrac%7B1%7D%7B2%2B%5Cfrac%7B1%7D%7Bn%7D%7D%3D%5Cfrac%7B1%7D%7B2%2B0%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="\lim_{n\rightarrow\infty}S_n=\lim_{n\rightarrow\infty}\frac{n}{2n+1}=\lim_{n\rightarrow\infty}\frac{n}{n\left(2+\frac{1}{n}\right)}=\lim_{n\rightarrow\infty}=\frac{1}{2+\frac{1}{n}}=\frac{1}{2+0}=\frac{1}{2}"/><br/> <div>Sarjan sumam on 1/2.</div> <div> </div> <div>Lause:</div> <div>Jos sarja <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7Da_k" alt="\sum_{k=1}^{\infty}a_k"/>suppenee, niin <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bk%5Crightarrow%5Cinfty%7Da_k%3D0" alt="\lim_{k\rightarrow\infty}a_k=0"/></div> <div> </div> <div>Huom:</div> <div>Jos <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bk%5Crightarrow%5Cinfty%7Da_k%3D0" alt="\lim_{k\rightarrow\infty}a_k=0"/>, niin sarja voi supeta tai hajaantua</div> <div>Jos <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bk%5Crightarrow%5Cinfty%7Da_k%5Cne0" alt="\lim_{k\rightarrow\infty}a_k\ne0"/>tai lukujonolla <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(a_k%5Cright)" alt="\left(a_k\right)"/>ei ole raja-arvoa, sarja hajaantuu.</div> <div> </div> <div>Esimerkki. Suppeneeko vai hajaantuuko sarja?</div> <div>a) </div> <img src="https://math-demo.abitti.fi/math.svg?latex=1-1%2B%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B1%7D%7B4%7D-%5Cfrac%7B1%7D%7B4%7D%2B..." alt="1-1+\frac{1}{2}-\frac{1}{2}+\frac{1}{4}-\frac{1}{4}+..."/> <div>Lasketaan osasummin</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_1%3D1" alt="S_1=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=S_2%3D1-1%3D0" alt="S_2=1-1=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_3%3D1-1%2B%5Cfrac%7B1%7D%7B2%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="S_3=1-1+\frac{1}{2}=\frac{1}{2}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_4%3D1-1%2B%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B2%7D%3D0" alt="S_4=1-1+\frac{1}{2}-\frac{1}{2}=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_5%3D1-1%2B%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B1%7D%7B4%7D%3D%5Cfrac%7B1%7D%7B4%7D%5C%20" alt="S_5=1-1+\frac{1}{2}-\frac{1}{2}+\frac{1}{4}=\frac{1}{4}\ "/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_6%3D1-1%2B%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B1%7D%7B4%7D-%5Cfrac%7B1%7D%7B4%7D%3D0" alt="S_6=1-1+\frac{1}{2}-\frac{1}{2}+\frac{1}{4}-\frac{1}{4}=0"/> jne...</div> <div>Joka toinen osasumma on nolla ja joka toinen pienenee, joten sarja suppenee ja <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7DS_n%3D0" alt="\lim_{n\rightarrow\infty}S_n=0"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7D%5Cfrac%7B1%7D%7Bk%7D" alt="\sum_{k=1}^{\infty}\frac{1}{k}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bk%5Crightarrow%5Cinfty%7D%5Cfrac%7B1%7D%7Bk%7D%3D0" alt="\lim_{k\rightarrow\infty}\frac{1}{k}=0"/>eli sarja voi supeta tai hajaantua.</div> <div>Osasummat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_1%3D1" alt="S_1=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=S_2%3D1%2B%5Cfrac%7B1%7D%7B2%7D%3D%5Cfrac%7B3%7D%7B2%7D" alt="S_2=1+\frac{1}{2}=\frac{3}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=S_3%3D1%2B%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B1%7D%7B3%7D%3D%5Cfrac%7B11%7D%7B6%7D" alt="S_3=1+\frac{1}{2}+\frac{1}{3}=\frac{11}{6}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=S_4%3D1%2B%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B1%7D%7B3%7D%2B%5Cfrac%7B1%7D%7B4%7D%3D%5Cfrac%7B25%7D%7B12%7D" alt="S_4=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}=\frac{25}{12}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=S_5%3D%5Cfrac%7B137%7D%7B60%7D" alt="S_5=\frac{137}{60}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7DS_n%3D%5Cinfty" alt="\lim_{n\rightarrow\infty}S_n=\infty"/>eli sarja hajaantuu<br/> <div> </div> <div>Lause </div> <div>Geometrinen sarja suppenee joss peräkkäisten yhteenlaskettavien lukujen suhteelle q ≠ 0 pätee -1 &lt; q &lt; 1 (tai |q|&lt;1).</div> <div>Sarjan summa on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7Da_k%3D%5Cfrac%7Ba_1%7D%7B1-q%7D" alt="\sum_{k=1}^{\infty}a_k=\frac{a_1}{1-q}"/></div> <div>Muutoin geometrinen sarja hajaantuu.</div> <div> </div> <div>267</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%7B%2C%7D2121...%5C%20%3D3%2B0%7B%2C%7D2121%3D3%2B0%7B%2C%7D21%2B0%7B%2C%7D0021%2B..." alt="3{,}2121...\ =3+0{,}2121=3+0{,}21+0{,}0021+..."/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D3%2B21%5Ccdot0%7B%2C%7D01%2B21%5Ccdot0%7B%2C%7D0001%2B..." alt="=3+21\cdot0{,}01+21\cdot0{,}0001+..."/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D3%2B21%5Ccdot0%7B%2C%7D01%2B21%5Ccdot0%7B%2C%7D01%5E2%2B..." alt="=3+21\cdot0{,}01+21\cdot0{,}01^2+..."/></div> <div>Sarja <img src="https://math-demo.abitti.fi/math.svg?latex=21%5Ccdot0%7B%2C%7D01%2B21%5Ccdot0%7B%2C%7D01%5E2%2B..." alt="21\cdot0{,}01+21\cdot0{,}01^2+..."/>on geometrinen, jossa</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D0%7B%2C%7D21%5C%20ja%5C%20q%3D0%7B%2C%7D01" alt="a_1=0{,}21\ ja\ q=0{,}01"/></div> <div>Koska -1 &lt; q &lt; 1, sarja suppenee. Sarjan summa on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7Ba_1%7D%7B1-q%7D%3D%5Cfrac%7B0%7B%2C%7D21%7D%7B1-0%7B%2C%7D01%7D%3D%5Cfrac%7B0%7B%2C%7D21%7D%7B0%7B%2C%7D99%7D%3D%5Cfrac%7B21%7D%7B99%7D" alt="S=\frac{a_1}{1-q}=\frac{0{,}21}{1-0{,}01}=\frac{0{,}21}{0{,}99}=\frac{21}{99}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%7B%2C%7D2121...%3D3%2B%5Cfrac%7B21%7D%7B99%7D%3D%5Cfrac%7B318%7D%7B99%7D%5E%7B%5Ctext%7B(%7D3%7D%3D%5Cfrac%7B106%7D%7B33%7D" alt="3{,}2121...=3+\frac{21}{99}=\frac{318}{99}^{\text{(}3}=\frac{106}{33}"/></div> 2019-11-27T09:14:51+02:00 https://peda.net/id/2ef1ca5c0f5 2019-11-25T09:21:22+02:00 <div>Lukujono on geometrinen, jos peräkkäiste jäsenten suhde <img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D" alt="q=\frac{a_{n+1}}{a_n}"/>on vakio kaikilal n=1, 2, 3, ...</div> <div>Geometerinen lukujono on muotoa<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(a_1%7B%2C%7D%5C%20a_1%7B%2C%7D%5C%20a_1q%5E2%7B%2C%7D...%5Cright)" alt="\left(a_1{,}\ a_1{,}\ a_1q^2{,}...\right)"/>, missä <img src="https://math-demo.abitti.fi/math.svg?latex=a_1%5Cne0" alt="a_1\ne0"/>ja <img src="https://math-demo.abitti.fi/math.svg?latex=q%5Cne0" alt="q\ne0"/></div> <div>n:s jäsen on <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3Da_1q%5E%7Bn-1%7D" alt="a_n=a_1q^{n-1}"/></div> <div>Toisaalta n:s jäsen voidaan esittää muodossa </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3Da_1q%5E%7Bn-1%7D%3Da_1q%5Enq%5E%7B-1%7D%3Da_1q%5En%5Cfrac%7B1%7D%7Bq%7D%3D%5Cfrac%7Ba_1%7D%7Bq%7Dq%5En%3Dbq%5En" alt="a_n=a_1q^{n-1}=a_1q^nq^{-1}=a_1q^n\frac{1}{q}=\frac{a_1}{q}q^n=bq^n"/>, missä <img src="https://math-demo.abitti.fi/math.svg?latex=b%3D%5Cfrac%7Ba_1%7D%7Bq%7D%5Cleft(%5Cne0%5Cright)" alt="b=\frac{a_1}{q}\left(\ne0\right)"/></div> <div>Lause </div> <div>Geometrinen lukujono suppenee jos peräkkäisten jäsenten suhde on välillä -1 ≤ q ≤ 1.</div> <div>Tällöin</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7Da_n%3D0%7B%2C%7D%5C%20kun%5C%20-1%3Cq%3C1" alt="\lim_{n\rightarrow\infty}a_n=0{,}\ kun\ -1&lt;q&lt;1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7Da_n%3Da_1%7B%2C%7D%5C%20kun%5C%20q%3D1" alt="\lim_{n\rightarrow\infty}a_n=a_1{,}\ kun\ q=1"/> <div><br/> <div>Esimerkki. Onko lukujono gerometrinen? Määritä lukujonon raja-arvo. Suppeneeko lukujono?</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D5%5Ccdot1%7B%2C%7D25%5E%7Bn-1%7D" alt="a_n=5\cdot1{,}25^{n-1}"/></div> <div>Lukujonon n:sjäsen on mutoa <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3Da_1q%5E%7Bn-1%7D" alt="a_n=a_1q^{n-1}"/>, missä <img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D5%7B%2C%7D%5C%20q%3D1%7B%2C%7D25" alt="a_1=5{,}\ q=1{,}25"/>, joten lukujono on geomterinen.</div> <div>Tai:</div> <div>Lasketaan peräkkäisten jäsenten suhde</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B5%5Ccdot1%7B%2C%7D25%5E%7B%5Cleft(n%2B1%5Cright)-1%7D%7D%7B5%5Ccdot1%7B%2C%7D25%5E%7Bn-1%7D%7D%3D%5Cfrac%7B1%7B%2C%7D25%5En%7D%7B1%7B%2C%7D25%5E%7Bn-1%7D%7D%3D1%7B%2C%7D25%5E%7Bn-%5Cleft(n-1%5Cright)%7D%3D1%7B%2C%7D25" alt="\frac{a_{n+1}}{a_n}=\frac{5\cdot1{,}25^{\left(n+1\right)-1}}{5\cdot1{,}25^{n-1}}=\frac{1{,}25^n}{1{,}25^{n-1}}=1{,}25^{n-\left(n-1\right)}=1{,}25"/> vakio, joten lukujono on geometrinen lukujono.</div> <div> </div> <div>Koska peräkkäisten jäsenten suhde q=1,25&gt;1, lukujono hajaantuu. <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7D5%5Ccdot1%7B%2C%7D25%5E%7Bn-1%7D%3D%5Cinfty" alt="\lim_{n\rightarrow\infty}5\cdot1{,}25^{n-1}=\infty"/>(raja-arvoa ei ole)</div> <div> </div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B4%5Ccdot2%5E%7Bn%2B1%7D%7D%7B3%5En%7D%3D%5Cfrac%7B4%5Ccdot2%5En%5Ccdot2%5E1%7D%7B3%5En%7D%3D%5Cfrac%7B8%5Ccdot2%5En%7D%7B3%5En%7D%3D8%5Ccdot%5Cleft(%5Cfrac%7B2%7D%7B3%7D%5Cright)%5En" alt="a_n=\frac{4\cdot2^{n+1}}{3^n}=\frac{4\cdot2^n\cdot2^1}{3^n}=\frac{8\cdot2^n}{3^n}=8\cdot\left(\frac{2}{3}\right)^n"/></div> <div>Lukujonon n:s jäsen on muotoa <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3Dbq%5En%7B%2C%7D%5C%20miss%C3%A4%5C%20q%3D%5Cfrac%7B2%7D%7B3%7D%7B%2C%7D%5C%20b%3D%5Cfrac%7Ba_1%7D%7Bq%7D%3D8%7B%2C%7D%5C%20a_1%3Dbq%3D8%5Ccdot%5Cfrac%7B2%7D%7B3%7D%3D%5Cfrac%7B16%7D%7B3%7D" alt="a_n=bq^n{,}\ missä\ q=\frac{2}{3}{,}\ b=\frac{a_1}{q}=8{,}\ a_1=bq=8\cdot\frac{2}{3}=\frac{16}{3}"/></div> <div>eli lukujono on geometrinen</div> <div>Koska q=2/3, niin q∈]-1,1], joten lukujono suppenee ja <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7Da_n%3D0" alt="\lim_{n\rightarrow\infty}a_n=0"/></div> <br/> <br/> </div> 2019-11-25T09:21:22+02:00 https://peda.net/id/9c40393606c 2019-11-18T08:48:50+02:00 <div> </div> <div>Kun x→∞, muuttujan arvo suurenee rajatta</div> <div>Kun x→-∞, muuttujan arvo pienee rajatta</div> <div> </div> <div>Jos funktion raja-arvo äärettömyydessä on b, merkitään</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7Df%5Cleft(x%5Cright)%3Db%5C%20tai%5C%20f%5Cleft(x%5Cright)%5Crightarrow%20b%7B%2C%7D%5C%20kun%5C%20x%5Crightarrow%5Cinfty" alt="\lim_{x\rightarrow\infty}f\left(x\right)=b\ tai\ f\left(x\right)\rightarrow b{,}\ kun\ x\rightarrow\infty"/></div> <div>Jos funktion raja-arvo miinus äärettömyydessä on b, merkitään <br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-%5Cinfty%7Df%5Cleft(x%5Cright)%3Db%5C%20tai%5C%20f%5Cleft(x%5Cright)%5Crightarrow%20b%7B%2C%7D%5C%20kun%5C%20x%5Crightarrow-%5Cinfty" alt="\lim_{x\rightarrow-\infty}f\left(x\right)=b\ tai\ f\left(x\right)\rightarrow b{,}\ kun\ x\rightarrow-\infty"/></div> </div> <div> </div> <div>Lause</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B1%7D%7Bx%5En%7D%3D0%5C%20ja%5C%20%5Clim_%7Bx%5Crightarrow-%5Cinfty%7D%5Cfrac%7B1%7D%7Bx%5En%7D%3D0%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D_%2B" alt="\lim_{x\rightarrow\infty}\frac{1}{x^n}=0\ ja\ \lim_{x\rightarrow-\infty}\frac{1}{x^n}=0{,}\ n\in\mathbb{Z}_+"/></div> <div> </div> Esimerkki. Määritä <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B3x%5E3%2B5%7D%7Bx%5E3%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(%5Cfrac%7B3x%5E3%7D%7Bx%5E3%7D%2B%5Cfrac%7B5%7D%7Bx%5E3%7D%5Cright)%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(3%2B5%5Ccdot%5Cfrac%7B1%7D%7Bx%5E3%7D%5Cright)%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(3%2B5%5Ccdot0%5Cright)%3D3" alt="\lim_{x\rightarrow\infty}\frac{3x^3+5}{x^3}=\lim_{x\rightarrow\infty}\left(\frac{3x^3}{x^3}+\frac{5}{x^3}\right)=\lim_{x\rightarrow\infty}\left(3+5\cdot\frac{1}{x^3}\right)=\lim_{x\rightarrow\infty}\left(3+5\cdot0\right)=3"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-%5Cinfty%7D%5Cleft(x%5E4%2B2x%5E3%5Cright)%3D%5Clim_%7Bx%5Crightarrow-%5Cinfty%7D%5Cleft(x%5E4%5Cleft(1%2B%5Cfrac%7B2%7D%7Bx%7D%5Cright)%5Cright)" alt="\lim_{x\rightarrow-\infty}\left(x^4+2x^3\right)=\lim_{x\rightarrow-\infty}\left(x^4\left(1+\frac{2}{x}\right)\right)"/></div> Tässä<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7Bx%7D%5Crightarrow0%7B%2C%7D%5C%20kun%5C%20x%5Crightarrow-%5Cinfty" alt="\frac{2}{x}\rightarrow0{,}\ kun\ x\rightarrow-\infty"/>. Siis<img src="https://math-demo.abitti.fi/math.svg?latex=1%2B%5Cfrac%7B2%7D%7Bx%7D%5Crightarrow1%7B%2C%7D%5C%20kun%5C%20x%5Crightarrow-%5Cinfty" alt="1+\frac{2}{x}\rightarrow1{,}\ kun\ x\rightarrow-\infty"/></div> <div>Lisäksi<img src="https://math-demo.abitti.fi/math.svg?latex=x%5E4%5Crightarrow%5Cinfty%7B%2C%7D%5C%20kun%5C%20x%5Crightarrow-%5Cinfty" alt="x^4\rightarrow\infty{,}\ kun\ x\rightarrow-\infty"/></div> <div>Näin ollen<img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-%5Cinfty%7D%5Cleft(x%5E4%2B2x%5E3%5Cright)%3D%5Cinfty" alt="\lim_{x\rightarrow-\infty}\left(x^4+2x^3\right)=\infty"/></div> c) <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B4x%5E2%2B2x%7D%7Bx%5E3%2B2x%2B1%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Bx%5E3%5Cleft(%5Cfrac%7B4%7D%7Bx%7D%2B%5Cfrac%7B2%7D%7Bx%5E2%7D%5Cright)%7D%7Bx%5E3%5Cleft(1%2B%5Cfrac%7B2%7D%7Bx%5E2%7D%2B%5Cfrac%7B1%7D%7Bx%5E3%7D%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(%5Cfrac%7B4%5Ccdot%5Cfrac%7B1%7D%7Bx%7D%2B2%5Ccdot%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%7B1%2B2%5Ccdot%5Cfrac%7B1%7D%7Bx%5E2%7D%2B%5Cfrac%7B1%7D%7Bx%5E3%7D%7D%5Cright)%5Crightarrow%5Cfrac%7B4%5Ccdot0%2B2%5Ccdot0%7D%7B1%2B2%5Ccdot0%2B0%7D%3D%5Cfrac%7B0%7D%7B1%7D%3D0" alt="\lim_{x\rightarrow\infty}\frac{4x^2+2x}{x^3+2x+1}=\lim_{x\rightarrow\infty}\frac{x^3\left(\frac{4}{x}+\frac{2}{x^2}\right)}{x^3\left(1+\frac{2}{x^2}+\frac{1}{x^3}\right)}=\lim_{x\rightarrow\infty}\left(\frac{4\cdot\frac{1}{x}+2\cdot\frac{1}{x^2}}{1+2\cdot\frac{1}{x^2}+\frac{1}{x^3}}\right)\rightarrow\frac{4\cdot0+2\cdot0}{1+2\cdot0+0}=\frac{0}{1}=0"/><br/> Siis</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B4x%5E2%2B2x%7D%7Bx%5E3%2B2x%2B1%7D%3D0" alt="\lim_{x\rightarrow\infty}\frac{4x^2+2x}{x^3+2x+1}=0"/><br/> <br/> Lause <div> Kun a &gt; 1, niin </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7Da%5Ex%3D%5Cinfty%5C%20ja%5C%20%5Clim_%7Bx%5Crightarrow-%5Cinfty%7Da%5Ex%3D0" alt="\lim_{x\rightarrow\infty}a^x=\infty\ ja\ \lim_{x\rightarrow-\infty}a^x=0"/></div> Kun 0 &lt; a &lt; 1, niin <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7Da%5Ex%3D0%5C%20ja%5C%20%5Clim_%7Bx%5Crightarrow-%5Cinfty%7Da%5Ex%3D%5Cinfty" alt="\lim_{x\rightarrow\infty}a^x=0\ ja\ \lim_{x\rightarrow-\infty}a^x=\infty"/><br/> <br/> <div>Esimerkki:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B3%2B2e%5Ex%7D%7B2e%5Ex%7D" alt="\lim_{x\rightarrow\infty}\frac{3+2e^x}{2e^x}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%2B2e%5Ex%7D%7B2e%5Ex%7D%3D%5Cfrac%7B3%7D%7B2e%5Ex%7D%2B%5Cfrac%7B2e%5Ex%7D%7B2e%5Ex%7D%3D%5Cfrac%7B3%7D%7B2%7D%5Ccdot%5Cfrac%7B1%7D%7Be%5Ex%7D%2B1" alt="\frac{3+2e^x}{2e^x}=\frac{3}{2e^x}+\frac{2e^x}{2e^x}=\frac{3}{2}\cdot\frac{1}{e^x}+1"/></div> <div>e &gt; 1, joten <img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%5Crightarrow%5Cinfty" alt="e^x\rightarrow\infty"/>, kun <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Crightarrow%5Cinfty" alt="x\rightarrow\infty"/>. Siten</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7Be%5Ex%5C%20%7D%5Crightarrow0%7B%2C%7D%5C%20kun%5C%20x%5Crightarrow%5Cinfty" alt="\frac{1}{e^x\ }\rightarrow0{,}\ kun\ x\rightarrow\infty"/>. Siten </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B3%2B2e%5Ex%7D%7B2e%5Ex%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(%5Cfrac%7B3%7D%7B2%7D%5Ccdot%5Cfrac%7B1%7D%7Be%5Ex%7D%2B1%5Cright)%3D%5Cfrac%7B3%7D%7B2%7D%5Ccdot0%2B1%3D1" alt="\lim_{x\rightarrow\infty}\frac{3+2e^x}{2e^x}=\lim_{x\rightarrow\infty}\left(\frac{3}{2}\cdot\frac{1}{e^x}+1\right)=\frac{3}{2}\cdot0+1=1"/></div> <br/> <div> <div> </div> </div> </div> 2019-11-14T11:32:00+02:00