   Chapter 2.3, Problem 54E

Chapter
Section
Textbook Problem

Let f ( x ) = 〚 cos x 〛 ,   − π ≤ x ≤ π . (a) Sketch the graph of f.(b) Evaluate each limit, if it exists.(i) lim x → 0 f ( x ) (ii) lim x → ( π / 2 ) − f ( x ) (iii) lim x → ( π / 2 ) + f ( x ) (iv) lim x → ( π / 2 ) f ( x ) (c) For what values of a does lim x→0 f(x) exist ?

(a)

To determine

To sketch: The graph of f(x)=cosx where πxπ.

Explanation

Definition used:

Greatest integer function:

The largest integer that is less than or equal to x is, x.

Let f(x)=y and draw the function y=cosx on the interval πxπ as shown below in Figure 1.

Plug x=π in f(x)=cosx.

f(π)=cos(π)=cos(π)=1=1

Plug x=π2 in f(x)=cosx.

f(π2)=cos(π2)=cos(π2)=0=0

From Figure 1, it is observed that 1cosx<0 on the interval [π,π2).

By the definition of the greatest integer function, f(x)=cosx is −1 on the interval [π,π2).

Plug x=π2 in f(x)=cosx.

f(π2)=cos(π2)=cos(π2)=0=0

Plug x=0 in f(x)=cosx.

f(0)=cos(0)=1=1

Plug x=π2 in f(x)=cosx

(b)

To determine

To evaluate: The limit of the greatest integer function.

(c)

To determine

To find: The values of a does limxaf(x) exist.

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