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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 ) = { cos x if x < 0 0 if x = 0 1 x 2 if x > 0 a = 0

To determine

To explain: The function f(x)={cosxif x<10if x=01x2if x>0 is discontinuous at the number a=0 and sketch the graph of the function f(x)={cosxif x<10if x=01x2if x>0.

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”.

Theorem: The limit limxaf(x)=L if and only if limxaf(x)=L=limxa+f(x).

Calculation:

By note 1, the function f is said to be discontinuous at x=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)={cosxif x<10if x=01x2if x>0.

Here, the function f(x) is defined at x=a. That is, f(0)=0 is defined.

The limit of the function f(x) as x approaches a=0 is computed as follows

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