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Chapter 3: Q. 17 (page 310)

Calculate each of the limits in Exercises 15–20 (a) using L’Hôpital’s rule and (b) without using L’Hôpital’s rule.

17.limx→∞x-12-3x2

Short Answer

Expert verified

The value of the limit is0

Step by step solution

01

Part (a) Step 1. Given information

We have been given the limit aslimx→∞x-12-3x2

02

Part (a) Step 2. Find the limit using L’Hôpital’s rule.

limx→∞x-12-3x2=limx→∞1-6x=0

03

Part (b) Step 1. Find the limit without using L’Hôpital’s rule.

limx→∞x-12-3x2=limx→∞1x-1x22x2-3=limx→∞0-00-3=0-3=0

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Most popular questions from this chapter

Determine whether or not each function f in Exercises 53–60 satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of the Mean Value Theorem.

f(x)=x2−1x2−4,[a,b]=[−3,3]

Last night at 6 p.m., Linda got up from her blue easy chair. She did not return to her easy chair until she sat down again at 8 p.m. Let s(t) be the distance between Linda and her easy chair t minutes after 6 p.m. last night.

(a) Sketch a possible graph of s(t), and describe what Linda did between 6 p.m. and 8 p.m. according to your graph. (Questions to think about: Will Linda necessarily move in a continuous and differentiable way? What are good ranges for t and s?

(b) Use Rolle’s Theorem to show that at some point between 6 p.m. and 8 p.m., Linda’s velocity v(t) with respect to the easy chair was zero. Find such a place on the graph of s(t).

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

f(x)=(x-1)2x+2

Determine whether or not each function f in Exercises 53–60 satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of the Mean Value Theorem.

f(x)=ln(x2+1),[a,b]=[0,1]

Use the second-derivative test to determine the local extrema of each function fin Exercises 29-40. If the second-derivative test fails, you may use the first-derivative test. Then verify your algebraic answers with graphs from a calculator or graphing utility. (Note: These are the same functions that you examined with the first-derivative test in Exercises 39-50of Section 3.2.)

f(x)=(x-2)2(1+x)
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