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Chapter 3: Q53. (page 311)

Calculate each of the limits in Exercises 49鈥�64. Some of these limits are made easier by considering the logarithm of the limit first, and some are not.

limx鈫�1x1x-1

Short Answer

Expert verified

The required limit is e.

Step by step solution

01

Step 1. Given information.

Consider the given information,

limx鈫�1x1x-1

02

Step 2. Taking log on both sides.

Apply the logarithm property to simplify.

limx鈫�1+鈥�ln鈦�x1x-1=limx鈫�1+鈥�1(x鈭�1)ln鈦�xsince,log(a)b=blog(a)=limx鈫�1+鈥�ln鈦�x(x鈭�1)(00form)=limx鈫�1+鈥�1x(L'hospitalrule)=limx鈫�1+鈥�1x=1
03

Step 3. Calculating the value of the limit.

Find the limit from the above concepts,

limx鈫�1+鈥�x1(x鈭�1)=e1=e

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Most popular questions from this chapter

Restate Theorem 3.3 so that its conclusion has to do with

tangent lines.

Sketch careful, labeled graphs of each function f in Exercises 63鈥�82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f,f',andf'', and examine any relevant limits so that you can describe all key points and behaviors of f.

f(x)=x2+3x

Find the critical points of the function

f'(x)=(x-1)(x-2)x-3

Find the critical points of f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

fx=21-Inx

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.
See all solutions

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