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Chapter 1: Q. 65 (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, $1$ .

Step by step solution

01

Step 1. Given Information.

The limit is,

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

02

Step 2. Finding the solution.

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

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

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

Inputting $h = 0$ ,

$= 0^{2} + 3 (0) + 1 = 0 + 0 + 1 = 1$

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Most popular questions from this chapter

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\lim_{x \to - 5} (3 x - 2) .$

Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .

$\lim_{x \to - 6} (x + 2) = - 4$ .

For each limit statement , use algebra to find δ > 0 in terms of $ε$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $ε$ .

$\lim_{x \to - 2} (4 - 2 x) = 8$

For each limit statement in Exercises $41 - 44$ , use algebra to find $δ$ or $N$ in terms of $ε$ or $M$ , according to the appropriate formal limit definition.

$\lim_{x \to \infty} (x^{2} + 2) = \infty$ ,find $N$ in terms of $M$ .

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f (x) = \{\begin{array}{l} (x - 3), i f x < 0 \\ - (x - 3), i f x ⩾ 0 \end{array} .$

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