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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

Explain why the function is discontinuous at the given number a. Sketch the graph of the function.

f ( x ) = { 1 x + 2 if x 2 1 if x = 2 a = 2

To determine

To explain: The function f(x)={1x+2if x21if x=2 is discontinuous at the number a=2 and sketch the graph of the function f(x)={1x+2if x21if x=2.

Explanation

Definition used: “A function f is continuous at a number a if limxaf(x)=f(a)”.

Note 1: “If f is defined near a, f is discontinuous at a whenever f is not continuous at a”.

Calculation:

By note 1, the function f is said to be discontinuous at a if anyone of the following conditions does not satisfied.

  • f(a) is defined
  • The limit of the function at the number a exists.
  • limxaf(x)=f(a)

Consider the piecewise function f(x)={1x+2if x21if x=2.

Here, f(2)=1 is defined.

The limit of the function f(x) as x approaches a=2 but x2 is computed as follows.

Consider the left hand limit limx21x+2

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