Chapter 1: Q. 66 (page 136)

Calculate each of the limits:
$\lim_{h \to 0} \frac{{(- 1 + h)}^{3} - {(- 1)}^{3}}{h}$ .

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information.

The limit is,

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

Step 2. Finding the solution.

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

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

$= \lim_{h \to 0} \frac{h^{3} - 3 h^{2} + 3 h - 1 - (- 1)}{h} = \lim_{h \to 0} \frac{h^{3} - 3 h^{2} + 3 h - 1 + 1}{h} = \lim_{h \to 0} \frac{h^{3} - 3 h^{2} + 3 h}{h} = \lim_{h \to 0} (h^{2} - 3 h + 3)$

Inputting $h = 0$ ,

$= {(0)}^{2} - 3 (0) + 3 = 0 - 0 + 3 = 3$

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Calculate each of the limits:
$\lim_{h \to 0} \frac{{(- 1 + h)}^{3} - {(- 1)}^{3}}{h}$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the solution.

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Calculate each of the limits:limh→0(-1+h)3-(-1)3h.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the solution.

One App. One Place for Learning.

Most popular questions from this chapter

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Calculate each of the limits:
$\lim_{h \to 0} \frac{{(- 1 + h)}^{3} - {(- 1)}^{3}}{h}$ .