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Chapter 1: Q. 90 (page 137)

Use limit rules and the continuity of polynomial functions to prove that every rational function is continuous on its domain.

Short Answer

Expert verified

A rational function r=pqis continuous at every point where q鈮�0.

Step by step solution

01

Step 1. Given Information:

Using limit rules and the continuity of power functions.

02

Step 2. Prove:

Consider any constant function,

f(x)=1

Again, the identity function g(x)=xare continuous on 鈩�

03

Step 3. Every rational function is continuous in its domain 

Now scalar multiples, sums, and products imply that every polynomial is continuous on鈩�

It also follows that a rational function r=pqis continuous at every point whereq鈮�0 .

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Most popular questions from this chapter

For each limit limx鈫�cf(x)=Lin Exercises 43鈥�54, use graphs and algebra to approximate the largest value of 未such that ifx鈭�(c-未,c)鈭�(c,c+未)thenf(x)鈭�(L-蔚,L+蔚)

limx鈫�-1x2-2x-3x+1=-4,蔚=0.1

For each limit statement , use algebra to find 未 > 0 in terms of 蔚> 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < 蔚.

limx鈫�0(x3+1)=1

Calculate each of the limits:

limx鈫�3xtan-1x

Explain why the Intermediate Value Theorem allows us to say that a function can change sign only at discontinuities and zeroes.

Write delta-epsilon proofs for each of the limit statements in Exercises 47鈥�60.

limx鈫�2x2-3x+2x-2=1
See all solutions

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