Q. 63 - Calculus

Question

Calculate each of the limits:

$\underset{h\to 0}{\lim }\frac{{(3+h)}^2-3^2}{h}$.

Step-by-Step Solution

Answer

The solution for the limit $\underset{h\to 0}{\lim }\frac{{(3+h)}^2-3^2}{h}$ is, $6$.

Step 1. Given Information.

The limit is,

$\underset{h\to 0}{\lim }\frac{{(3+h)}^2-3^2}{h}$.

Step 2. Finding the solution

$\underset{h\to 0}{\lim }\frac{{(3+h)}^2-3^2}{h}$

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

$$\begin{gathered} =\underset{h\to 0}{\lim }\frac{9+6h+h^2-9}{h} \\ =\underset{h\to 0}{\lim }\frac{h^2+6h}{h} \\ =\underset{h\to 0}{\lim }h+6 \end{gathered}$$

Inputting $h=0$,

$$\begin{gathered} =0+6 \\ =6 \end{gathered}$$