Chapter 3: Q. 38 (page 311)

Calculate each of the limits in Exercises $21 - 48$ . Some of these limits are made easier by L’Hopital’s rule, and some are not.
$\lim_{x \to 2^{+}} \ln (\frac{2^{x} - 4}{x - 2})$ .

Short Answer

Expert verified

The value of the limit $\lim_{x \to 2^{+}} \ln (\frac{2^{x} - 4}{x - 2})$ is, $\ln (\ln 16)$ .

Step by step solution

Step 1. Given information

$\lim_{x \to 2^{+}} \ln (\frac{2^{x} - 4}{x - 2})$ .

Step 2. The limit using L'Hopital's rule is given below:

$\lim_{x \to 2^{+}} \ln (\frac{2^{x} - 4}{x - 2}) = \lim_{x \to 2^{+}} \ln (\frac{2^{x} - 4}{x - 2}) = \ln [\lim_{x \to 2^{+}} (\frac{2^{x} - 4}{x - 2})]$

$= \ln [\lim_{x \to 2^{+}} (\frac{2^{x} \ln 2}{1 - 0})]$ [L'Hopital's rule]

$= \ln [\lim_{x \to 2^{+}} (2^{x} \ln 2)] = \ln (2^{2} \ln 2) = \ln (4 \ln 2) = \ln (\ln 2^{4}) = \ln (\ln 16)$

The exact value is, $\ln (\ln 16)$ .

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Calculate each of the limits in Exercises $21 - 48$ . Some of these limits are made easier by L’Hopital’s rule, and some are not.
$\lim_{x \to 2^{+}} \ln (\frac{2^{x} - 4}{x - 2})$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. The limit using L'Hopital's rule is given below:

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Calculate each of the limits in Exercises 21-48. Some of these limits are made easier by L’Hopital’s rule, and some are not.limx→2+ln2x-4x-2.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The limit using L'Hopital's rule is given below:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Probability and Statistics

Mechanics Maths

Decision Maths

Statistics

Study anywhere. Anytime. Across all devices.

Calculate each of the limits in Exercises $21 - 48$ . Some of these limits are made easier by L’Hopital’s rule, and some are not.
$\lim_{x \to 2^{+}} \ln (\frac{2^{x} - 4}{x - 2})$ .