Chapter 3: Q. 33 (page 311)

Calculate each of the limits in Exercises 21-48. Some of these limits are made easier by L'Hôpital's rule, and some are not.
$\lim_{x \to \infty} \frac{x e^{x}}{e^{2 x} + 1}$

Short Answer

Expert verified

The value of limit is zero

Step by step solution

Step 1. Given information

Given function $\lim_{x \to \infty} \frac{x e^{x}}{e^{2 x} + 1}$

Apply L-Hospital rule and solve

Calculating, we get

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

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Calculate each of the limits in Exercises 21-48. Some of these limits are made easier by L'Hôpital's rule, and some are not.
$\lim_{x \to \infty} \frac{x e^{x}}{e^{2 x} + 1}$

Short Answer

Step by step solution

Step 1. Given information

Apply L-Hospital rule and solve

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Calculate each of the limits in Exercises 21-48. Some of these limits are made easier by L'Hôpital's rule, and some are not.limx→∞xexe2x+1

Short Answer

Step by step solution

Step 1. Given information

Apply L-Hospital rule and solve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

Logic and Functions

Probability and Statistics

Mechanics 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'Hôpital's rule, and some are not.
$\lim_{x \to \infty} \frac{x e^{x}}{e^{2 x} + 1}$