CPM Homework Help : CALC Problem 10-142

Calculate the following limits.

$\lim\limits_ { x \rightarrow 0 ^ { + } } x ^ { 2 } \operatorname { cot } x$
$\lim\limits_ { x \rightarrow 0 } ( e ^ { x } + x ) ^ { 1 / x }$
Hint: Use natural log.
$\lim\limits_ { x \rightarrow \infty } \frac { e ^ { x } } { x ^ { 2 } }$
$\lim\limits_ { x \rightarrow \infty } \frac { x } { \sqrt { x ^ { 2 } + 1 } }$

Step (a):

Rewrite this limit as:

$\lim_{x\to 0^+}\frac{x^2}{\tan(x)}$

Then l'Hôpital's Rule can be used.

Step 1 (b):

$L=\lim_{x\to 0}(e^x+x)^{1/x}$

$\ln(L)=\ln\Big(\lim_{x\to 0}(e^x+x)^{1/x}\Big)$

Step 2 (b):

$\ln(L)=\lim_{x\to 0}\frac{1}{x}\ln(e^x+x)=\lim_{x\to 0}\frac{\ln(e^x+x)}{x}$

Now l'Hôpital's Rule can be used.

More Help (b):

When the limit is evaluated, the value is ln(L). Be sure to determine the value of L.