   Chapter 3.4, Problem 44E

Chapter
Section
Textbook Problem

# Make a rough sketch of the curve y = x n (n an integer) for the following five cases:(i) n = 0 (ii) n > 0 , n  odd (iii) n > 0 , n  even (iv) n < 0 , n  odd (v) n < 0 , n  even Then use these sketches to find the following limits.(a) lim x →   0 + x n (b) lim x → 0 − x n (c) lim x →   ∞ x n (d) lim x → − ∞ x n

To determine

To:

1) Make a rough sketch of the curve y=xn for the following five cases

(i) n=0

(ii) n>0, n odd

(iii) n>0, n even

(iv) n<0, n odd

(v) n<0, n even

2) Find the value of limits

(a) limx0+xn

(b) limx0-xn

(c) limxxn

(d) limx-xn

Explanation

1) Concept:

First draw the rough sketches and from that find the value of limits

2) Calculation:

A) Rough sketches are as follows

(i) For n=0,y=x0=1. Therefore, the sketch is

(ii) For n>0, n odd y=xn. Therefore,

(iii) For n>0, n even y=xn. Therefore, the sketch is

(iv) For n<0, n odd y=x-n. Therefore, the sketch is

(v) For n<0, n even y=x-n. Therefore, the sketch is

B) Values of limits are,

(a) limx0+xn

At n=0.From the sketch limx0+xn=1

At n>0, From the sketch limx0+xn=0

At n<0, From the sketch limx0+xn=

(b) limx0-xn

At n=0. From the sketch limx0-xn=1

At n>0. From the sketch limx0-xn=0

At n<0, n odd. From the sketch limx0-xn=-

At n<0, n even. From the sketch limx0-xn=

(c) limxxn

At n=0. From the sketch limxxn=1

At n>0. From the sketch limxxn=

At n<0, From the sketch limxxn=0

(d) limx-xn

At n=0

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