\subsubsection {}
Spoètìte následující limity pomocí Taylorova
vzorce: 


 ${\lim\limits_{x\to 0}}
{e^x \sin x - x(1+x)\over x^3} \quad 


{\lim\limits_{x\to 0}}
\left(\frac1x-{1\over\sin x}\right)\quad

{\lim\limits_{x\to   {+\infty}   }}\left(x-x^2\ln\left(1+\frac1x\right)\right)$ 




\subsubsection {}
1.  Najdìte $n\in\Bbb N$, aby limita

a) ${\lim\limits_{x\to 0}}{\cos x-\cos(\tg x)\over x^n}$, \quad

b) ${\lim\limits_{x\to 0}}{e-(1+x)^{\frac1x}\over x^n}$
byla koneèná a rùzná od $0$, a spoètìte tuto limitu.

  2.  Najdìte $a,b\in\Bbb R$, aby ${\lim\limits_{x\to 0}} {x-a\sin
x-b\tg x\over x^4}=0$ a spoètìte ${\lim\limits_{x\to 0}} {x-a\sin
x-b\tg x\over x^5}$.




\subsubsection {}
1. ${\lim\limits_{x\to 0}} {\cos x- e^{-x^2\over 2}\over x^4}$,\quad

{2.} $${\lim\limits_{x\to   {+\infty}   }} \left(\root 6 \of
{x^6+x^5} - \root 6 \of {x^6-x^5}\right),$

 ${3.} $${\lim\limits_{x\to   {+\infty}   }} x^{\frac32}\left(\sqrt{x+1}+\sqrt{x-1}-2\sqrt
x\right),$

 ${4.} ${\lim\limits_{x\to 0}} {a^x+a^{-x}-2\over x^2} \ (a>0)$,\quad

{5.} ${\lim\limits_{x\to 0}} \frac1x\left(\frac1x -{\cos x\over \sin
x}\right)$,\quad

{6.} $${\lim\limits_{x\to   {+\infty}   }}\left(\left(x^3-x^2+\frac
x2\right)e^{\frac1x}-\sqrt{x^6+1}\right).$

  7.  Najdìte $n\in\Bbb N$, aby limita

a) ${\lim\limits_{x\to 0}}{\tg(\sin x)-\sin(\tg x)\over x^n}$, \quad

b) ${\lim\limits_{x\to 0}}{(1+x)^x-1\over x^n}$
byla koneèná a rùzná od $0$, a spoètìte tuto limitu.

  8.  Najdìte $a,b\in\Bbb R$, aby ${\lim\limits_{x\to 0}} {x-(a+b\cos
x)\sin x\over x^4}=0$.


