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Chapter 1: Q. 14 (page 148)

Describe in terms of large and small numbers why it makes intuitive sense that limits of the forma鈭�+鈭�(b)鈭�路鈭�(c)鈭�1must be infinite.

Short Answer

Expert verified

(a) The limit 鈭�+鈭�is infinity and is a large number.

(b) The limit 鈭�路鈭�is infinity and is a large number.

(c) The limit 鈭�1 is infinity and is a large number.

Step by step solution

01

Step 1. Given Information

The given limits are鈭�+鈭�,鈭�路鈭�and鈭�1

02

Part (a) Step 1. The limit ∞+∞

The limit of the form 鈭�+鈭�is non-indeterminate form and it gives a very large number.

03

Part(b) Step 1. The limit ∞·∞

The infinity multiplied by infinity is infinity which is a large number.

04

Part (c) Step 1. The limit ∞1

鈭�1=鈭� because any number raised power 1 is itself the number.

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Most popular questions from this chapter

For each limit statement , use algebra to find 未 > 0 in terms of 蔚> 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < 蔚.

limx鈫�21x=12;youmayassume未鈮�1

For each limit statement , use algebra to find 未 > 0 in terms of 蔚> 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < 蔚.

limx鈫�0(3-4x)=3

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鈫�cis the value fc.

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

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

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

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

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

(h) Iflimx鈫�鈭�fx=2, then the graph offhas a horizontal asymptote aty=2.

Use algebra to find the largest possible value of 未 or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.
Ifx>N, then1鈭�2x<鈭�500

For each limit statement , use algebra to find 未 > 0 in terms of 蔚> 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < 蔚.

limx鈫�3(x2-6x+5)=-4
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