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6-53.

Evaluate. Homework Help ✎

1. $\lim\limits _ { n \rightarrow 0 } \frac { \int _ { 2 } ^ { 2 + n } ( \sqrt { x ^ { 3 } + 1 } ) d x } { n }$

Be careful! Evaluating this limit as n → 0 leads to trouble:

$\lim_{n\rightarrow 0}\frac{\int_{2}^{2+0} \left(\sqrt{x^{3}+1}\right)dx}{n}=\frac{0}{0}\text{ Recall that }\frac{0}{0}\text{ is an indeterminate form.}$

That means we do not know if the limit is finite (because the 0s 'cancel out') or infinite (because there is a 0 in the denominator).

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:

$f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$

Since n → 0, notice that a = 0. So we can rewrite the limit as:

$f'(0)=\lim_{n\rightarrow 0}\frac{\int_{2}^{2+0}\left(\sqrt{x^{3}+1}\right)dx-0}{n-0}$

$\text{This means that }f(x)=\int_{2}^{2+n}\left(\sqrt{x^{3}+1}\right)dx.$

So, we just need to find f '(x) at x = 0 and we have evaluated the limit.

$\text{By the Fundamental Theorem of Calculus: }\frac{d}{dx}\int_{2}^{2+n}\left (\sqrt{x^{3}+1} \right )dx=\sqrt{x^{3}+1}$

Now evaluate that derivative at x = 0:

$f'(0)=\sqrt{(0)^{3}+1}=1$

1. $\lim\limits _ { x \rightarrow \pi } \frac { \int _ { \pi } ^ { x } \operatorname { cos } ( t ^ { 2 } ) d t } { x - \pi }$

Refer to part (a).

1. $\lim\limits _ { x \rightarrow - \infty } \frac { x ^ { 2 } } { e ^ { - x } }$

$\lim_{n\rightarrow -\infty}\frac{x^2}{e^{-x}}=\frac{\infty}{\infty}$

l’Hôpital’s Rule can be used.

1. $\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ( x ^ { 2 } ) } { x \operatorname { tan } ( x ) }$

See part (c).
Be sure to reevaluate the limit after each use of l’Hôpital’s Rule.