### Home > APCALC > Chapter 4 > Lesson 4.1.2 > Problem4-24

4-24.

Evaluate the following limits.

1. $\lim \limits _ { x \rightarrow 2 } \frac { x ^ { 2 } - 4 } { x - 2 }$

Factor first. The denominator might 'cancel out.'

2. $\lim \limits _ { h \rightarrow 0 } \frac { ( 1 + h ) ^ { 2 } - 1 } { h }$

This is Hana's definition of a derivative at a point.

$f'(a)=\lim \limits_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}$

What is $f\left(x\right)$?
What is $a$?
What is $f^\prime\left(x\right)$?
What is $f^\prime\left(a\right)$?

$f\left(x\right) = x^{2}$
$a = 1$
$f ^\prime\left(x\right) = 2x$
$f ^\prime\left(a\right) = 2$
Therefore the limit as $h→0 = f^\prime \left(a\right) = 2$