Chapter 9.2, Problem 8E

### 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 3-8, determine whether each function is continuous or discontinuous at the given x-value. Examine the three conditions in the definition of continuity. f ( x ) = { x 2 + 1  if  x ≤ 1 2 x 2 − 1  if  x > 1 , x = 1

To determine

Whether the function, f(x)={x2+1      if x12x21   if x>1, is continuous or discontinuous at x=1.

Explanation

Given Information:

The provided function is f(x)={x2+1Â Â Â Â Â Â ifÂ xâ‰¤12x2âˆ’1Â Â Â ifÂ x>1.

Explanation:

Consider the provided function,

f(x)={x2+1Â Â Â Â Â Â ifÂ xâ‰¤12x2âˆ’1Â Â Â ifÂ x>1

A function, f, is continuous at x=c if f(c) exists, limxâ†’cf(x) exists and limxâ†’cf(x)=f(c).

The function is continuous at x=1 if all the three conditions are satisfied for x=1.

To check whether the value of f(1) exists, substitute 1 for x in the function,

f(1)=12+1=1+1=2

Thus, the value of f(1) is defined, which means the first condition is satisfied, and is equal to 2.

To check whether the limit, limxâ†’1f(x), exists, find the values of the left hand limit and the right hand limit. If the values of the left hand limit and right hand limit are equal, then the limit exists.

The limit from the left is represented by limxâ†’1âˆ’f(x), that is, the values of f(1) but c<1

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