  ### Home > APCALC > Chapter 8 > Lesson 8.3.2 > Problem8-119

8-119.

Evaluate the following limits.

1. $\lim\limits_ { x \rightarrow 6 } \frac { \operatorname { ln } ( x - 4 ) - \operatorname { ln } ( 2 ) } { x - 6 }$

This is Ana's Definition of the Derivative.

1. $\lim\limits _ { x \rightarrow 25 } \frac { \sqrt { 2 x - 1 } - 7 } { x - 25 }$

See the hint in part (a).

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

For limits to infinity, we can compare the numerator with the denominator. As the numerator approaches infinity, $y = \sin^2(x)$ 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.

$=\lim \limits_{x\rightarrow \infty }\frac{\text{small}}{\text{big}}=0$

1. $\lim\limits _ { x \rightarrow \infty } \frac { 3 ^ { x } + 7 x } { e ^ { x } + 10 x }$

Recall that the number $e$ is about $2.718$. It is larger than $2$ and smaller than $3$. Use that to compare the dominant terms of the numerator and denominator.