Chapter 1: Q. 1 (page 119)

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
$f (x) = \{\begin{aligned} \frac{1}{x} \sin (x), if x \neq 0 \\ 1, if x = 0 \end{aligned}$

Short Answer

Expert verified

Function is 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)=1

$\begin{aligned} lim_{x \to a^{-}} f (x) = f (a) \\ lim_{x \to a^{+}} f (x) = f (a) \end{aligned}$

Left continuity

role="math" localid="1649854552566" $lim_{x \to a^{-}} f (x) = f (a)$

Right continuity

$\begin{aligned} lim_{x \to a^{+}} f (x) = f (a) \\ lim_{x \to 0^{-}} f (x) = lim_{x \to 0^{-}} x \sin (\frac{1}{x}) \end{aligned}$

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

Step 3. Solving for right continuity

$\begin{aligned} lim_{x \to 0^{-}} f (x) = lim_{x \to 0^{-}} \frac{\frac{x}{x} \sin (\frac{1}{x})}{(\frac{1}{x})} \\ lim_{x \to 0^{-}} f (x) = lim_{x \to 0^{-}} 1 \\ lim_{x \to 0^{-}} f (x) = 1 \\ lim_{x \to 0^{+}} f (x) = lim_{x \to 0^{+}} x \sin (\frac{1}{x}) \end{aligned}$

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

$\begin{aligned} lim_{x \to 0^{+}} f (x) = lim_{x \to 0^{+}} \frac{\frac{x}{x} \sin (\frac{1}{x})}{(\frac{1}{x})} \\ lim_{x \to 0^{+}} f (x) = lim_{x \to 0^{+}} 1 \\ lim_{x \to 0^{+}} f (x) = 1 \\ lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = f (a) \end{aligned}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

Step 3. Solving for right continuity

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Geometry

Applied Mathematics

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.

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)=1xsin⁡(x),ifx≠01,ifx=0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

Step 3. Solving for right continuity

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Geometry

Applied Mathematics

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.