# How to find these limits?

• Dec 23rd 2011, 02:23 AM
gotmejerry
How to find these limits?
Hello! Today was my first calculus exam, and I couldn't solve these to problems. Can you help me with these?

$\lim_{x\to\0}\frac{1-\sqrt[3]{\cos{x}}}{1-\cos{\sqrt[3]{x}}}$

$\lim_{n\to\infty}n\cdot(\sqrt[n]{2011}-1)$

And is this convergent, why or why not?
$\sum_{k=1}^{\infty}\frac{1}{\sqrt{k}}\sin^2\frac{1 }{\sqrt{k}}$

Thank you, and happy holidays!
• Dec 23rd 2011, 02:37 AM
odx
Re: How to find these limits?
$\lim_{n \to \infty }n(\sqrt[n]{2011}-1)=\lim_{n \to \infty }\frac{2011^{\frac{1}{n}}-1}{\frac{1}{n}}=ln 2011$
• Dec 23rd 2011, 02:58 AM
CaptainBlack
Re: How to find these limits?
Quote:

Originally Posted by gotmejerry
And is this convergent, why or why not?
$\sum_{k=1}^{\infty}\frac{1}{\sqrt{k}}\sin^2\frac{1 }{\sqrt{k}}$

Since:

$\frac{1}{\sqrt{k}}\sin^2\left(\frac{1}{\sqrt{k}} \right)=O(k^{-3/2})$

the series converges.

CB
• Dec 23rd 2011, 03:51 AM
sbhatnagar
Re: How to find these limits?
$\lim_{x \to 0}\frac{1-\sqrt[3]{\cos{x}}}{1-\cos{\sqrt[3]{x}}}$

$\frac{0}{0 }$ indeterminate form

\begin{align*} \lim_{x \to 0}\frac{1-\sqrt[3]{\cos{x}}}{1-\cos{\sqrt[3]{x}}} &= \lim_{x \to 0} \frac{x^{2 \over 3}\sin{x}}{\sin(\sqrt[3]{x})\cos(x^{2 \over 3})} \quad [\text{L 'Hospital's rule}]\\ \\ &= -\lim_{x \to 0} \frac{\sqrt[3]{n\cdot \cos{x}}[2\sin{x}+3x\cos{x}]}{2x^{2\over 3}\sin{\sqrt[3]{x}}\sin{x}-\cos{\sqrt[3]{x}}\cos{x}} \quad [\text{L 'Hospital's rule}] \\ \\ &= -\frac{0}{-1} \\ \\&=0\end{align*}