Chapter 3: Q. 51 (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.
$\lim_{x \to 2^{+}} (x - 2)^{x^{2} - 4}$ .

Short Answer

Expert verified

The exact value of the limit $\lim_{x \to 2^{+}} (x - 2)^{x^{2} - 4}$ is, $1 .$

Step by step solution

Step 1. Given information

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

Step 2. Taking logarithm of the limit.

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

$= \lim_{x \to 2^{+}} \frac{\ln (x - 2)}{(x^{2} - 4)}$ [ in the form of $\frac{\infty}{\infty}$ ]

$= \lim_{x \to 2^{+}} \frac{\frac{1}{(x - 2)}}{- \frac{1}{{(x^{2} - 4)}^{2}} \cdot 2 x} = - \lim_{x \to 2^{+}} \frac{1}{(x - 2)} \frac{{(x^{2} - 4)}^{2}}{2 x} = - \lim_{x \to 2^{+}} \frac{1}{(x - 2)} \frac{(x - 2)^{2} (x + 2)^{2}}{2 x} = - \lim_{x \to 2^{+}} \frac{(x - 2) (x + 2)^{2}}{2 x} = 0$

Step 3. Therefore, the value of the limit is given by,

$\lim_{x \to 2^{+}} \ln (x - 2)^{x^{2} - 4} = e^{0} = 1$

Therefore, the exact value is, $1 .$

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Taking logarithm of the limit.

Step 3. Therefore, the value of the limit is given by,

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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→2+(x-2)x2-4.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Taking logarithm of the limit.

Step 3. Therefore, the value of the limit is given by,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Calculus

Pure Maths

Logic and Functions

Applied Mathematics

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