Chapter 3: Q. 59 (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.
$\lim_{x \to 1 +} {(x - 1)}^{\ln x}$

Short Answer

Expert verified

$\lim_{x \to 1 +} {(x - 1)}^{\ln x} = 1$

Step by step solution

Step 1. Given information

$\lim_{x \to 1 +} {(x - 1)}^{\ln x}$

Step 2. Taking log on both sides

$\begin{aligned} lim_{x \to 1^{+}} \ln (x - 1)^{\ln x} = lim_{x \to 1^{+}} (\ln x) \ln (x - 1) \\ = lim_{x \to 1^{+}} \frac{\ln (x - 1)}{\frac{1}{\ln x}} [in the form of \frac{\infty}{\infty}] \\ = lim_{x \to 1^{+}} (\frac{\frac{1}{x - 1}}{- (\ln x)^{- 2} \cdot \frac{1}{x}}) [using L'Hospitals rule] \\ = lim_{x \to 1^{+}} (\frac{\frac{1}{x - 1}}{- \frac{1}{x (\ln x)^{2}}}) \end{aligned}$

role="math" localid="1649429584354" $\begin{aligned} = - lim_{x \to 1^{+}} (\frac{1}{x - 1} \cdot x (\ln x)^{2}) \\ = - lim_{x \to 1^{+}} (\frac{x (\ln x)^{2}}{x - 1}) [in the form of \frac{\infty}{\infty}] \\ = - lim_{x \to 1^{+}} (\frac{x \cdot 2 (\ln x) \cdot \frac{1}{x} + (\ln x)^{2} \cdot 1}{1}) \\ = - lim_{x \to 1^{+}} (2 (\ln x) + (\ln x)^{2}) \\ = - 2 \ln 1 - (\ln 1)^{2} \\ = 0 - 0 \\ = 0 \end{aligned}$

Step 3. Calculating the value of limit

$\begin{aligned} lim_{x \to 1^{+}} (x - 1)^{\ln x} = e^{0} \\ = 1 \end{aligned}$

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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.
$\lim_{x \to 1 +} {(x - 1)}^{\ln x}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Taking log on both sides

Step 3. Calculating the value of limit

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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→1+(x−1)lnx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Taking log on both sides

Step 3. Calculating the value of limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Probability and Statistics

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.
$\lim_{x \to 1 +} {(x - 1)}^{\ln x}$