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6-53.

Evaluate.

$\lim\limits_ { n \rightarrow 0 } \frac { \int _ { 2 } ^ { 2 + n } ( \sqrt { x ^ { 3 } + 1 } ) d x } { n }$
Hint (a):
Be careful! Evaluating this limit as $n → 0$ leads us to trouble:
$\lim\limits_{n\rightarrow 0}\frac{\int_{2}^{2+0} \left(\sqrt{x^{3}+1}\right)dx}{n}=\frac{0}{0}$ Recall that $\frac{0}{0}$ is an indeterminate form.
That means we do not know if the limit is finite (because the $0$ 's 'cancel out') or infinite (because there is a $0$ in the denominator).
Steps (a):
We can evaluate an indeterminate limit if that limit happens to be a definition of the derivative.
Sure enough, part (a) is Ana's Definition of the Derivative:
Since $n → 0$ , notice that $a = 0$ . So we can rewrite the limit as:
So, we just need to find $f^\prime(x)$ at $x = 0$ and we have evaluated the limit.
Now evaluate that derivative at $x = 0$ :

$\lim\limits_ { x \rightarrow \pi } \frac { \int _ { \pi } ^ { x } \operatorname { cos }( t ^ { 2 }) d t } { x - \pi }$
Hint (b):
Refer to part (a).

$\lim\limits_ { x \rightarrow - \infty } \frac { x ^ { 2 } } { e ^ { - x } }$

$\lim\limits_ { x \rightarrow 0 } \frac { \operatorname { sin } ( x ^ { 2 } ) } { x \operatorname { tan } ( x ) }$

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