   Chapter 2.5, Problem 71E

Chapter
Section
Textbook Problem

Show that the function f ( x ) = { x 4 sin ( 1 / x )   if   x ≠ 0 0   if   x = 0 is continuous on ( –∞, ∞).

To determine

To show: The function f(x)={x4sin(1x)if   x00ifx=0

is continuous in (,), that is if limx0f(x)=0=f(0).

Explanation

Given:

f(x)={x4sin(1x)if   x00ifx=0

Formula used:

(1) Squeeze Theorem: If limx0f(x)=0=limx0g(x) and f(x)h(x)g(x) then limx0h(x)=0.

The f(x)=x4sin(1x) is continues on (,0)(0,), since it is product of polynomial and a.

Composite of a trigonometric function and a rational function and the above three function is

continuous in the interval (,0)(0,) because 0 is not included.

Only need to show that the function f(x)=x4sin(1x) is continues at x=0

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