   Chapter 2, Problem 25RE

Chapter
Section
Textbook Problem

Sketch the graph of the functionf(x) = { 2 x − 3  if   x    ≤   2 − x  + 3    if   x   >   2 and evaluate lim x → a + f(x), lim x → a − f(x), and lim x → a f(x) at the point a = 2, if the limits exist.

To determine

To sketch: The graph of the given function and evaluate limxa+f(x) , limxaf(x) and limxaf(x) at the point a=2 , if limit exists.

Explanation

Given:

The given function is,

f(x)={2x3ifx2x+3ifx>2 (1)

Calculation:

Substitute different values of x and find the corresponding values to the function respectively.

Substitute 0 for x in equation (1) to find the value of f(x) ,

f(0)=203=03=3

Substitute 1 for x in equation (1) to find the value of f(x) ,

f(1)=213=23=1

Substitute 1.5 for x in equation (1) to find the value of f(x) ,

f(1.5)=2(1.5)3=33=0

Substitute 2 for x in equation (1) to find the value of f(x) ,

f(2)=223=43=1

Substitute 3 for x in equation (1) to find the value of f(x) ,

f(3)=3+3=0

Substitute 4 for x in equation (1) to find the value of f(x) ,

f(4)=4+3=1

Substitute 5 for x in equation (1) to find the value of f(x) ,

f(5)=5+3=2

From above calculation substitute different values of x and f(x) in a table which is given below,

 x f(x) 0 −3 1 −1 1

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