Q. 65 - Calculus

Question

Calculate each of the limits:

$\underset{h\to 0}{\lim }\frac{{(1+h)}^3-1^3}{h}$.

Step-by-Step Solution

Answer

The solution for the limit $\underset{h\to 0}{\lim }\frac{{(1+h)}^3-1^3}{h}$ is, $1$.

Step 1. Given Information.

The limit is,

$\underset{h\to 0}{\lim }\frac{{(1+h)}^3-1^3}{h}$.

Step 2. Finding the solution.

$\underset{h\to 0}{\lim }\frac{{(1+h)}^3-1^3}{h}$

Using ${(a+b)}^3=a^3+3a^{2}b+3a{b}^2+b^3$,

$$\begin{gathered} =\underset{h\to 0}{\lim }\frac{1+h^3+3h+3h^2-1}{h} \\ =\underset{h\to 0}{\lim }\frac{h^3+3h^2+h}{h} \\ =\underset{h\to 0}{\lim }(h^2+3h+1) \end{gathered}$$

Inputting $h=0$,

$$\begin{gathered} =0^2+3\left(0\right)+1 \\ =0+0+1 \\ =1 \end{gathered}$$