   Chapter 9, Problem 30RE Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Solutions

Chapter
Section Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

In Problems 25-30, suppose that f ( x ) = { x 2 + 1       if  x ≤ 1 x               if 0 < x < 1 2 x 2 − 1     if  x ≥ 1 Is f ( x ) continuous at x = − 1 ?

To determine

Whether the function, f(x)={x2+1      if x0x           if 0<x<12x21   if x1 is continuous at x=1.

Explanation

Given Information:

The function is f(x)={x2+1      if x0x           if 0<x<12x21   if x1.

Explanation:

Consider the provided function,

f(x)={x2+1      if x0x           if 0<x<12x21   if x1

A function f(x) is continuous at x=c if the limit limxcf(x) exists. So, to check whether the function is continuous at x=1, consider the limit,

limx1f(x)

The limxcf(x) will exist at c if,

limxcf(x)=limxc+f(x)

The limit from the left is represented by limx1f(x) and the limit from the right is represented by limx1+f(x).

The limxcf(x) will exist at c=1 when the limit from the left, that is, the values of f(c) but c<1, is equal to the limit from the right, that is, the values of f(c) but c>1.

Consider the limit from the left,

limx1f(x)

As the f(x)=x2+1 for x<1

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