Chapter 3.1, Problem 1QR

To determine

To calculate: The value of the limit algebraically.

Answer to Problem 1QR

The required value of the limit is 4.

Explanation of Solution

Given information:

The limit $=\underset{h\to 0}{\lim }\frac{{\left(2+h\right)}^2-4}{h}$

Calculation:

It is seen that at $h=0$ ; the given limit is undefined.

So, to find the value of the limit eliminate the denominator.

Here, simplify the numerator of the above limit that is, ${\left(2+h\right)}^2-4$ .

  $\begin{array}{c}{\left(2+h\right)}^2-4=\cancel{4}+4h+h^2-\cancel{4}\\ =4h+h^2\end{array}$

Substitute $4h+h^2$ for ${\left(2+h\right)}^2-4$ in the given limit and simplify.

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

Hence, the required value of the limit is 4.