Chapter 3: Q. 32 (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.
$\lim_{x \to \infty} \frac{x e^{- 3 x}}{x^{2} + 3 x + 1}$

Short Answer

Expert verified

The value of limit is $0$

Step by step solution

Step 1. Given information

Given function $\lim_{x \to \infty} \frac{x e^{- 3 x}}{x^{2} + 3 x + 1}$

Apply L-Hospital rule and solve

Calculating, we get

$\lim_{x \to \infty} \frac{x e^{- 3 x}}{x^{2} + 3 x + 1} = \lim_{x \to \infty} \frac{x}{e^{3 x} (x^{2} + 3 x + 1)} \lim_{x \to \infty} \frac{x e^{- 3 x}}{x^{2} + 3 x + 1} = \lim_{x \to \infty} \frac{x}{e^{3 x} (x^{2} + 3 x + 1)} = \lim_{x \to \infty} \frac{1}{e^{3 x} (2 x + 3) + (x^{2} + 3 x + 1) \cdot e^{3 x} \cdot 3} = \lim_{x \to \infty} \frac{1}{e^{3 x} [2 x + 3 + 3 x^{2} + 9 x + 3]} = \lim_{x \to \infty} \frac{1}{e^{3 x} (3 x^{2} + 11 x + 6)} = 0$

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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.
$\lim_{x \to \infty} \frac{x e^{- 3 x}}{x^{2} + 3 x + 1}$

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→∞xe-3xx2+3x+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

Probability and Statistics

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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.
$\lim_{x \to \infty} \frac{x e^{- 3 x}}{x^{2} + 3 x + 1}$