Q. 2 Interesting trigonometric limits... [FREE SOLUTION]

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Chapter 1: Q. 2 (page 122)

Interesting trigonometric limits: For each of the functions that follow, use a calculator or other graphing utility to examine the graph of f near x = 0. Does it appear that f is continuous at x = 0? Make sure your calculator is set to radian mode
$f (x) = \{\begin{aligned} \sin 鈦� (\frac{1}{x}), if x 鈮� 0 \\ 0, if x = 0 \end{aligned}$

Short Answer

Expert verified

Function is not continuous at the given point.

Step by step solution

Step 1. Given information

We have to explain that f is continuous at x=0

Step 2. Explanation

Put x=0

f(0)=0

$\begin{aligned} lim_{x 鈫� a^{鈭�}} 鈥� f (x) = f (a) \\ lim_{x 鈫� a^{+}} 鈥� f (x) = f (a) \end{aligned}$

Left continuity

$lim_{x 鈫� a^{鈭�}} 鈥� f (x) = f (a)$

Right continuity

$\begin{aligned} lim_{x 鈫� a^{+}} 鈥� f (x) = f (a) \\ lim_{x 鈫� 0^{鈭�}} 鈥� f (x) = lim_{x 鈫� 0^{鈭�}} 鈥� x \sin 鈦� (\frac{1}{x}) \end{aligned}$

Dividing and multiplying by $\frac{1}{x}$

$\begin{aligned} lim_{x 鈫� 0^{鈭�}} 鈥� f (x) = lim_{x 鈫� 0^{鈭�}} 鈥� \frac{\frac{x}{x} \sin 鈦� (\frac{1}{x})}{(\frac{1}{x})} \\ lim_{x 鈫� 0^{鈭�}} 鈥� f (x) = lim_{x 鈫� 0^{鈭�}} 鈥� 1 \\ lim_{x 鈫� 0^{鈭�}} 鈥� f (x) = 1 \\ lim_{x 鈫� 0^{+}} 鈥� f (x) = lim_{x 鈫� 0^{+}} 鈥� x \sin 鈦� (\frac{1}{x}) \end{aligned}$

Dividing and multiplying by $\frac{1}{x}$

$\begin{aligned} lim_{x 鈫� 0^{+}} 鈥� f (x) = lim_{x 鈫� 0^{+}} 鈥� \frac{\frac{x}{x} \sin 鈦� (\frac{1}{x})}{(\frac{1}{x})} \\ lim_{x 鈫� 0^{+}} 鈥� f (x) = lim_{x 鈫� 0^{+}} 鈥� 1 \\ lim_{x 鈫� 0^{+}} 鈥� f (x) = 1 \\ lim_{x 鈫� a^{鈭�}} 鈥� f (x) = lim_{x 鈫� a^{+}} 鈥� f (x) 鈮� f (a) \end{aligned}$

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Interesting trigonometric limits: For each of the functions that follow, use a calculator or other graphing utility to examine the graph of f near x = 0. Does it appear that f is continuous at x = 0? Make sure your calculator is set to radian modef(x)=sin鈦�1x,ifx鈮�00,ifx=0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Theoretical and Mathematical Physics

Pure Maths

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.