  ### Home > APCALC > Chapter 4 > Lesson 4.4.1 > Problem4-141

4-141.

Evaluate each of the following limits.

1. $\lim\limits _ { x \rightarrow 9 } \frac { \sqrt { x } - 3 } { x - 9 }$

You could multiply the numerator and denominator by the conjugate of the numerator.

Or... you could recognize that this is Ana's Definition of the Derivative.

2. $\lim\limits _ { h \rightarrow 0 } \frac { \sqrt { 2 + h } - \sqrt { 2 } } { h }$

This is Hana's Definition of the Derivative.

What is $f\left(x\right)$?
What is $a$?
What is $f^\prime\left(x\right)$?
The $\text{limit} = f^\prime (a) = \text{_________}$

3. $\lim\limits _ { x \rightarrow \infty } \frac { 2 \sqrt { x } + 1 } { 5 - \sqrt { x } }$

This limit is approaching infinity. What is the end behavior? Is there a horizontal asymptote? Compare the highest-power terms in the numerator and denominator. Pay attention to coefficients.

$= −2$

4. $\lim\limits _ { x \rightarrow \infty } \operatorname { cos } ( x )$

Visualize the graph of $y=\cos x$. It oscillates as $x→∞$.