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Chapter 1: Q. 99 (page 137)

Use the quotient rule for limits and the continuity of cosxto prove that f(x)=secxis continuous on its domain.

Short Answer

Expert verified

It is proved that f(x)=secx is continuous on its domain.

Step by step solution

01

Step 1. Given Information

We are given thatcosx is continous.

02

Step 2. Proving the statement 

If x=cis in the domain of secx, then c is not a multiple of 蟺2, and cosc鈮�0. Therefore, limx鈫�csecx=limx鈫�c1cosxis the quotient of limx鈫�c1by,

limx鈫�ccosx=cosc

or

1cosc=secc

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Most popular questions from this chapter

Sketch the graph of a function f described in Exercises 27鈥�32, if possible. If it is not possible, explain why not.

f is left continuous at x = 1 and right continuous at x = 1, but is not continuous at x = 1, and f(1) = 鈭�2.

Calculate each of the limits:

limh鈫�0(2+h)2-22h.

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

limx鈫�-5x.

Write each of the inequalities in interval notation:

3x+1-2<0.1

For each functionf graphed in Exercises23鈥�26, describe the intervals on whichf is continuous. For each discontinuity off, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.

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