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Chapter 1: Q. 63 (page 136)

Calculate each of the limits:
$\lim_{h \to 0} \frac{{(3 + h)}^{2} - 3^{2}}{h}$ .

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given Information.

The limit is,

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

02

Step 2. Finding the solution

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

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

$= \lim_{h \to 0} \frac{9 + 6 h + h^{2} - 9}{h} = \lim_{h \to 0} \frac{h^{2} + 6 h}{h} = \lim_{h \to 0} h + 6$

Inputting $h = 0$ ,

$= 0 + 6 = 6$

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Most popular questions from this chapter

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

$\lim_{x \to 2^{-}} f (x) = - \infty, \lim_{x \to 2^{+}} f (x) = \infty, f (2) = 3 .$

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| < $ε$ .

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

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

$\lim_{x \to 0^{-}} f (x) = - 1, \lim_{x \to 0^{+}} f (x) = 1, f (0) = 0 .$

State what it means for a function f to be right continuous at a point x = c, in terms of the delta–epsilon definition of limit.

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