   Chapter 2.5, Problem 18E

Chapter
Section
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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