   Chapter 2.3, Problem 48E

Chapter
Section
Textbook Problem

Let g(x) =sgn(sinx).(a) Find each of the following limits or explain why it does not exist.(i) lim x → 0 + g ( x ) (ii) lim x → 0 − g ( x ) (iii) lim x → 0 g ( x ) (iv) lim x → π + g ( x ) (v) lim x → π − g ( x ) (vi) lim x → π g ( x ) (b) For which values of a does lim x → a g ( x ) not exist?(c) Sketch a graph of g.

(a)

To determine

To find: The value of the limit if exists.

Explanation

Theorem 1:

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

Given:

The function g(x)=sgn(sinx).

The signum function is defined as follows.

sgn(x)={1if x<00if x=01if x>0

Calculation:

Section (i)

Obtain the limit of the function g(x) as x approaches right side of 0.

Consider the smallest positive number near 0 is 0+.

Compute, limx0+sgn(sinx).

limx0+sgn(sinx)=sgnlimx0+(sinx)=sgnlimx0+(sin(0+))=sgn(0+)=1

Thus, the limit exists and limit value of the function is 1.

Section (ii)

Obtain the limit of the function g(x) as x approaches left side of 0.

From the definition, sgnx=1 when x<0.

Consider the smallest negative number near 0 is 0.

Compute, limx0sgn(sinx).

limx0sgn(sinx)=sgnlimx0(sinx)=sgnlimx0(sin(0))=sgn(0)=1

Thus, the limit exists and limit value of the function is 1_.

Section (iii)

Obtain the limit of the function g(x) as x approaches 0.

By Theorem 1, the limit of the function g(x) when x approaches zero exists if and only if the limit of the function g(x) when x approaches right-side of zero and the limit of the function g(x) when x approaches left side of zero are equal.

From Section (i) and Section (ii), limx0+sgn(sinx)=1 and limx0sgn(sinx)=1.

The limit of the function g(x) does not exist as the left and right-hand side limits are different

(b)

To determine

To find: The values of a when the limxag(x) not exist.

(c)

To determine

To sketch: The graph of the function,sgn(sinx)={1if sinx<00if sinx=01if sinx>0.

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