CPM Homework Help : APCALC Problem 10-160

Calculate each of the following limits.

$\lim\limits _ { x \rightarrow 0 ^ { + } } x ^ { 2 } \operatorname { cot } ( x )$
Step (a):
Rewrite this limit as:
$\lim\limits_{x\to 0^+}\frac{x^2}{\tan(x)}$
Then l'Hôpital's Rule can be used.

Hint: Use natural log.

Step 1 (b):

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

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

Step 2 (b):

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

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 $L$ .

$\lim\limits _ { x \rightarrow \infty } \frac { e ^ { x } } { x ^ { 2 } }$
Hint (c):
Which is the dominant function: $e^x$ or $x^2$ ?

$\lim\limits _ { x \rightarrow \infty } \frac { x } { \sqrt { x ^ { 2 } + 1 } }$
Hint (d):
What are the dominant terms in the numerator and denominator?