  ### Home > APCALC > Chapter 5 > Lesson 5.4.1 > Problem5-143

5-143.

Evaluate each of the following limits. Homework Help ✎

1. $\lim\limits _ { x \rightarrow 2 } \frac { x ^ { 3 } - 8 } { x - 2 }$

You could evaluate this limit algebraically.
Recall that: $x^3 − 8 = (x − 2)(x^2 + 2x − 4)$
Or you could recognize that this is Ana's Definition of the Derivative.
$f(x) = x^3$
$a = 2$
So the $\text{limit} = f '(2)$

2. $\lim\limits _ { h \rightarrow 0 } \frac { \operatorname { sec } ( 3 ( x + h ) ) - \operatorname { sec } ( 3 ( x - h ) ) } { 2 h }$

You should recognize that this is Hanah's Definition of the Derivative.

Deconstruct Hanah's definition.
1. Determine what is $f(x)$.
2. Find its derivative. That is the value of the limit.

3. $\lim\limits _ { h \rightarrow 0 } \frac { \operatorname { cot } ( \frac { \pi } { 4 } + h ) - 1 } { h }$

This is Hana's Definition of the Derivative, evaluated at

$x=\frac{\pi}{4}.$

4. $\lim\limits _ { x \rightarrow \pi } \frac { \operatorname { tan } ( 5 x ) } { x - \pi }$

This is Ana's Definition of the Derivative, evaluated at
$x = π$.

This limit could be rewritten as

$\lim_{x\rightarrow \pi}\frac{\text{tan}(5x)-\text{tan}(5\pi)}{x-\pi}$

Of course, $\operatorname{tan}(5π) = 0$.

5. What do all of the limits above have in common?

Refer to hints above.