F37 Taylorpolynom forts. Gränsvärden

Gränsvärden

$\underset{x\to a}{\lim }\frac{f(x)}{g(x)}$ $[\frac{0}{0}]$

Taylors formel

$f(x)=P_n(x)+E_n(x)$
där $P_n(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2}(x-a{)}^2+..+\frac{f^n(a)}{n!}(x-a{)}^n$

och $E_n(x)=\frac{f^{n+1}(s)}{(n+1)!}(x-a{)}^{n+1}$ för något $s$ mellan $a$ och $x$.

Kan skrivas $f(x)=P_n(x)+(x-a{)}^{n+1}B(x)$

Exempel

$\underset{x\to 0}{\lim }\frac{x^x-1}{x}$ $[\frac{0}{0}]$

$e^x=1+x+\frac{x^2}{2}+x^{3}B(x)$
$\underset{x\to 0}{\lim }\frac{e^x-1}{x}=\underset{x\to 0}{\lim }\frac{1+x+\frac{x^2}{2}+x^{3}B(x)-1}{x}$
$=\underset{x\to 0}{\lim }\frac{x+\frac{x^2}{2}+x^{3}B(x)}{x}=\underset{x\to 0}{\lim }1+\frac{x}{2}+x^{2}B(x)=1$

$\underset{x\to 0}{\lim }\frac{\sin (x)}{x}$ $[\frac{0}{0}]$

$f(x)=f(a)+f^{\prime }(a)(x-a)+x^{2}B(x)$
$\sin (x)=\sin (0)+\cos (0)(x-0)+x^{2}B(x)=x+x^{2}B(x)$
$\underset{x\to 0}{\lim }\frac{\sin (x)}{x}=\underset{x\to 0}{\lim }\frac{x+x^{2}B(x)}{x}=\underset{x\to 0}{\lim }1+xB(x)=1$

$\underset{x\to 1}{\lim }\frac{\ln x}{x^2-1}$ $[\frac{0}{0}]$

$f(x)=f(a)+f^{\prime }(a)(x-a)+f^{\prime \prime }(a)(x-a{)}^2+(x-a{)}^{3}B(x)$
$\ln x=\ln 1+\frac{1}{1}(x-1)+\frac{-1}{1^2}(x-1{)}^2+(x-1{)}^{3}B(x)$
$=(x-1)-(x-1{)}^2+(x-1{)}^{3}B(x)$ $\underset{x\to 1}{\lim }\frac{(x-1)-(x-1{)}^2+(x-1{)}^{3}B(x)}{x^2-1}=\underset{x\to 1}{\lim }\frac{(x-1)-(x-1{)}^2+(x-1{)}^{3}B(x)}{(x-1)(x+1)}$
$\underset{x\to 1}{\lim }\frac{1-(x-1)+(x-1{)}^{2}B(x)}{x+1}=\frac{1}{2}$

$\underset{x\to 0}{\lim }(\cos x{)}^{\frac{1}{x^2}}$

Notera $f(x)=(\cos x{)}^{\frac{1}{x^2}}>0$ för $x$ nära $0$
$e^{\ln f(x)}=e^{\ln {\cos x}^{\frac{1}{x^2}}}=e^{\frac{\ln (\cos x)}{x^2}}$ $[e^{[\frac{0}{0}]}]$
$\underset{x\to 0}{\lim }(\cos x{)}^{\frac{1}{x^2}}=\underset{x\to 0}{\lim }{e}^{\frac{\ln (\cos x)}{x^2}}=e^{\underset{x\to 0}{\lim }(\frac{\ln (\cos x)}{x^2})}$
Taylor utveckling kring $x=0$
låt $g(x)=\ln (\cos x)$ $g(0)=\ln (\cos 0)=\ln 1=0$
$g^{\prime }(x)=\frac{1}{\cos x}(-\sin x)=-\tan x$ $g^{\prime }(0)=-\frac{\sin 0}{\cos 0}=0$
$g^{\prime \prime }(x)=\frac{-1}{\cos x}$ $g^{\prime \prime }(0)=-\frac{1}{\cos 0}=-1$
Vi får $g(x)=0+0\cdot x+\frac{-1}{2}{x}^2+x^{3}B(x)$
$\underset{x\to 0}{\lim }\frac{\ln (\cos x)}{x^2}=\underset{x\to 0}{\lim }\frac{-\frac{x^2}{2}+x^{3}B(x)}{x^2}$
$=\underset{x\to 0}{\lim }-\frac{1}{2}+xB(x)=-\frac{1}{2}$
Vi får $\underset{x\to 0}{\lim }(\cos x{)}^{\frac{1}{x^2}}=e^{\underset{x\to 0}{\lim }(\frac{\ln (\cos x)}{x^2})}=e^{-\frac{1}{2}}=\frac{1}{\sqrt{(}e)}$