Question

Evaluate the limit if it exists.

$\underset{x\to 2}{\lim }\frac{x^4-16}{x-2}$

Short answer

The value of limit of the given function is$32.$

Step-by-step solution

Step 1. Given information.

The limit of function is given here,

$\underset{x\to 2}{\lim }\frac{x^4-16}{x-2}$

Step 2. Formula used.

The given function is a rational number.

We can find the limit by cancelling common factor.

We are using the formula:$a^2-b^2=(a+b)(a-b)$

Step 3. Simplification.

Given, $\underset{x\to 2}{\lim }\frac{x^4-16}{x-2}$

$\begin{array}{c}\underset{x\to 2}{\lim }\frac{x^4-16}{x-2}=\frac{{\left(x^2\right)}^2-{\left(4\right)}^2}{x-2}\\ =\frac{(x^2+4)(x^2-4)}{x-2}\\ =\frac{(x^2+4){\left(x\right)}^2-{\left(2\right)}^2}{x-2}\\ =\frac{(x^2+4)(x+2)(x-2)}{(x-2)}\\ =(x^2+4)(x+2)\end{array}$

Now, we put the value:

$\begin{array}{c}=(2^2+4)(2+2)\\ =(4+4)\left(4\right)\\ =8\times 4\\ =32\end{array}$

Therefore, our final answer is$32.$