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Chapter 1: Q. 5 TF (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

The function fis continuous atx=c.

Short Answer

Expert verified

The proof islimx鈫�x0(f(x)-f(c))=0

Step by step solution

01

Step 1. Given information

The function fis differentiable at x=cthen the function f is continuous atx=c.

02

Step 2. Calculation

A function is said to be continuous over a range if it is graph is a single and unbroken curve.

Formally,

A real valued function f(x)is said to be continuous at a point x=x0in domain if

limx鈫�x0f(x)exists and is equal to f(x0)

If a function f(x)is continuous at x=x0then

limx鈫�x0+f(x)=limx鈫�x0-f(x)=limx鈫�x0f(x)

Function that are not continuous are said to be discontinuous

Hence proved,

limx鈫�x0f(x)=limx鈫�cf(c)limx鈫�x0f(x)-limx鈫�cf(c)=0limx鈫�x0(f(x)-f(c))=0

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Most popular questions from this chapter

Sketch a labeled graph of a function that fails to satisfy the hypothesis of the Intermediate Value Theorem, and illustrate on your graph that the conclusion of the Intermediate Value Theorem does not necessarily hold.

For each function f graphed in Exercises 23鈥�26, describe the intervals on which f is continuous. For each discontinuity of f, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.

Use algebra to find the largest possible value of 未 or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.

Ifx>N, then1x2鈭�0<0.001

Each function in Exercises 9鈥�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.

limx鈫�2-f(x)=2,limx鈫�2+f(x)=1,f(2)=1.

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