18. One-sided and two-sided limits Use the graph of g in the figure to find the following values or state that they do notexist. If a limit does not exist, explain why.ina. g(2)b. lim g(x)x→2e. g(3)f. lim g(x)x 3g. lim g(x)x→3+c. lim g(x)x→2+yd. lim g(x)x→2h. g(4)i. lim g(x)x 4415320123y = g(x)45X

Answered: 18. One-sided and two-sided limits Use…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

18. One-sided and two-sided limits Use the graph of g in the figure to find the following values or state that they do not
exist. If a limit does not exist, explain why.
in
a. g(2)
b. lim g(x)
x→2
e. g(3)
f. lim g(x)
x 3
g. lim g(x)
x→3+
c. lim g(x)
x→2+
y
d. lim g(x)
x→2
h. g(4)
i. lim g(x)
x 4
4
1
5
3
2
0
1
2
3
y = g(x)
4
5
X

Expert Solution

Step 1

a. In the graph , corresponding to x = 2 , the point y = 3 is darkened inside which means the point y = 3 is included .

And the point y = 2 is not darkened hence it is not included

So g (2) = 3

b. $\underset{x \to 2^{-}}{\lim g (x)}$ is the left limit of g at x= 2.

To the left of x = 2 the graph is a straight line. As x = 2 is approached from left clearly g approaches 2 (but not attained)

So $\underset{x \to 2^{-}}{\lim g (x)}$ = 2

c. $\lim_{x \to 2^{+}} g (x)$ is the right limit of g at x= 2.

To the right of x = 2 , g(x) is constant when x < 3

So $\lim_{x \to 2^{+}} g (x)$ = 3

d. Since 2 = $\underset{x \to 2^{-}}{\lim g (x)}$ $\neq$ $\lim_{x \to 2^{+}} g (x)$ = 3 , $\lim_{x \to 2} g (x)$ does not exists

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18. One-sided and two-sided limits Use the graph of g in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. a. g(2) b. lim g(x) x-27 c. lim g(x) x→2+ d. lim g(x) e. g(3) f. lim g(x) x-3 g. lim g(x) 2-3+ h. g(4) i. lim g(x) x 4