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Chapter 1: Q. 60 (page 88)

Sketch graphs by hand and use them to make approximations for each of the limits in Exercises 53–66. If a two-sided limit does not exist, describe the one-sided limits.

limx→-∞3e4x+1

Short Answer

Expert verified

Graph for the given limit

The limit expression has the value oflimx→-∞3e4x+1=1

Step by step solution

01

Step 1. Given information

The given limitlimx→-∞3e4x+1

02

Step 2. Sketch graphs by hand and use them to make approximations for the limit

The graph is drawn as below:

From the graph,we see that forx→-∞ the function has a value off(x)=1

Therefore,the limit expression has the value ofrole="math" localid="1648547784846" limx→-∞3e4x+1=1

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Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of fxas x→cis the value fc.

(c) The limit of fxas x→cmight exist even if the value of fcdoes not.

(d) The two-sided limit of fxas x→cexists if and only if the left and right limits of fxexists as x→c.

(e) If the graph of fhas a vertical asymptote at x=5, then limx→5fx=∞.

(f) If limx→5fx=∞, then the graph of fhas a vertical asymptote at x=5.

(g) If limx→2fx=∞, then the graph of fhas a horizontal asymptote at x=2.

(h) Iflimx→∞fx=2, then the graph offhas a horizontal asymptote aty=2.
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