Q. 66 - 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,$3$

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

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

Inputting $h=0$,

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