Chapter 9, Problem 29RE

### Mathematical Applications for the ...

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

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 ≤ 0 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+1Â Â Â Â Â Â â€‰ifÂ xâ‰¤0xÂ Â Â Â Â Â Â Â â€‰Â Â Â â€‰ifÂ 0<x<12x2âˆ’1Â Â Â â€‰ifÂ xâ‰¥1.

Explanation:

Consider the provided function,

f(x)={x2+1Â Â Â Â Â Â â€‰ifÂ xâ‰¤0xÂ Â Â Â Â Â Â Â â€‰Â Â Â â€‰ifÂ 0<x<12x2âˆ’1Â Â Â â€‰ifÂ xâ‰¥1

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

limxâ†’1f(x)

The limxâ†’cf(x) will exist at c if,

limxâ†’câˆ’f(x)=limxâ†’c+f(x)

The limit from the left is represented by limxâ†’1âˆ’f(x) and the limit from the right is represented by limxâ†’1+f(x).

The limxâ†’cf(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

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