Chapter 3: Q. 58 (page 311)

Calculate each of the limits in Exercises 49–64. Some of these limits are made easier by considering the logarithm of the limit first, and some are not.
$\lim_{x \to \infty} {(\frac{1}{x - 1})}^{x}$

Short Answer

Expert verified

$\lim_{x \to \infty} {(\frac{1}{x - 1})}^{x} = 0$

Step by step solution

Step 1. Given information

$\lim_{x \to \infty} {(\frac{1}{x - 1})}^{x}$

Step 2. Taking log on both sides

$\begin{aligned} lim_{x \to \infty} {(\frac{1}{x - 1})}^{x} = lim_{x \to \infty} e^{\ln {(\frac{1}{x - 1})}^{x}} \\ = lim_{x \to \infty} e^{- x \ln (x - 1)} \\ = e^{- lim_{x \to \infty} x \ln (x - 1)} \\ = \frac{1}{e^{lim_{x \to \infty} x \ln (x - 1)}} \end{aligned}$

Step 3. Calculate limits

$lim_{x \to \infty} {(\frac{1}{x - 1})}^{x} = 0$

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Calculate each of the limits in Exercises 49–64. Some of these limits are made easier by considering the logarithm of the limit first, and some are not.
$\lim_{x \to \infty} {(\frac{1}{x - 1})}^{x}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Taking log on both sides

Step 3. Calculate limits

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Calculate each of the limits in Exercises 49–64. Some of these limits are made easier by considering the logarithm of the limit first, and some are not. limx→∞1x-1x

Short Answer

Step by step solution

Step 1. Given information

Step 2. Taking log on both sides

Step 3. Calculate limits

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Theoretical and Mathematical Physics

Logic and Functions

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

Calculate each of the limits in Exercises 49–64. Some of these limits are made easier by considering the logarithm of the limit first, and some are not.
$\lim_{x \to \infty} {(\frac{1}{x - 1})}^{x}$