Question

Evaluate the limit if it exists.

$\underset{h\to 0}{\lim }\frac{{(2+h)}^2-4}{h}$

Short answer

The value of limit of the given function is$0.$

Step-by-step solution

Step 1. Given information.

The limit of function is given here,

$\underset{h\to 0}{\lim }\frac{{(2+h)}^2-4}{h}$

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:${(a+b)}^2=a^2+b^2+2ab$

Step 3. Simplification.

Given, $\underset{h\to 0}{\lim }\frac{{(2+h)}^2-4}{h}$

$\begin{array}{c}\underset{h\to 0}{\lim }\frac{{(2+h)}^2-4}{h}=\frac{2^2+h^2+2\times 2\times h-4}{h}\\ =\frac{4+h^2+4h-4}{h}\\ =\frac{h^2+4h}{h}\\ =\frac{0+0}{0}\\ =0\end{array}$

Therefore, our final answer is$0.$