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7-197.

No calculator! Evaluate the following limits.

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ( x ^ { 2 } ) } { x ^ { 2 } }$
Hint (a):
$\text{This is an indeterminate form: }\frac{0}{0}.$
Use l'Hôpital's Rule.

$\lim\limits _ { x \rightarrow \infty } \frac { 100 x + 40 x ^ { 2 } - 5 x ^ { 3 } } { 10 x ^ { 3 } + 50 x ^ { 2 } - 100 }$
Hint (b):
This is a $\text{limit }→ ∞$ .
Compare the powers on the top and bottom only.
Do not neglect their coefficients.

$\lim\limits _ { x \rightarrow \pi ^ { + } } \operatorname { csc } ( x )$
Hint (c):
$\csc(x)=\frac{1}{\sin(x)}\text{ and }\sin(\pi) =0$
So this limit must be approaching $+∞$ or $−∞$ .
Which one?

$\lim\limits _ { h \rightarrow 0 } \frac { e ^ { 2 + h } - e ^ { 2 } } { h }$
Hint (d):
This is Hana's Definition of the Derivative.
The limit is equal to $f^\prime(2)$ when $f(x) = e^x$ .

$\lim\limits _ { x \rightarrow \infty } \frac { e ^ { x } + x ^ { 3 } } { e ^ { x } + x ^ { 2 } }$
Hint (e):
This is a $\text{limit }→ ∞$ . Be careful. Which is more powerful: an exponential function or a polynomial function?

Compute without a calculator

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