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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

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

f ( x ) = { 2 x 2 5 x 3 x 3 if x 3 6 if x = 2 a = 3

To determine

To explain: The function f(x)={2x25x3x3if x36if x=3 is discontinuous at the number a=3 and sketch the graph of the function f(x)={2x25x3x3if x36if x=3.

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”.

Fact 1:

If f(x)=g(x) when xa, then limxaf(x)=limxag(x), provided the limit exist.

Direct substitution property:

If f is a polynomial or a rational function and a is in the domain of f, then limxaf(x)=f(a).

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)={2x25x3x3if x36if x=3.

Here, the function f(3)=6 is defined.

The limit of the function f(x) as x approaches a=3 but x3 is computed as follows.

Simplify f(x) by using elementary algebra

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