CPM Homework Help : CALC Problem 8-118

Determine the following limits.

$\lim\limits_ { x \rightarrow 6 } \frac { \operatorname { ln } ( x - 4 ) - \operatorname { ln } ( 2 ) } { x - 6 }$
Hint (a):
This is Ana's Definition of the Derivative.

$\lim\limits_ { x \rightarrow 25 } \frac { \sqrt { 2 x - 1 } - 7 } { x - 25 }$
Hint (b):
See the hint in part (a).

$\lim\limits_ { x \rightarrow \infty } \frac { \operatorname { sin } ^ { 2 } x } { x }$
Hint (c):
For limits to infinity, we can compare the numerator with the denominator. As the numerator approaches infinity, $y =\operatorname{sin}^²x$ oscillates between $−1$ and $+1$ ... not very big. As the denominator approaches infinity, $y = x$ also approaches infinity. So the denominator is MUCH larger than the numerator.
Answer (c):
$=\lim\limits_{x\rightarrow \infty }\frac{\text{small}}{\text{big}}=0$

$\lim\limits_ { x \rightarrow \infty } \frac { 3 ^ { x } + 7 x } { e ^ { x } + 10 x }$
Hint (d):
Recall that the number $e$ is about $2.718$ . It's larger than $2$ and smaller than $3$ . Use that to compare the power of the numerator and denominator.