Chapter 1: Q. 43 (page 136)

Calculate each of the limits in Exercises $29 - 70$ .
$\lim_{x \to 1} \frac{x + x^{2} - 2 x^{3}}{x - x^{2}}$ .

Short Answer

Expert verified

The value of $\lim_{x \to 1} \frac{x + x^{2} - 2 x^{3}}{x - x^{2}}$ is, $3$ .

Step by step solution

Step 1. Given information

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

Step 2. The limit is calculated as below:

$\lim_{x \to 1} \frac{x + x^{2} - 2 x^{3}}{x - x^{2}} = \lim_{x \to 1} \frac{x (1 + x - 2 x^{2})}{x (1 - x)} = \lim_{x \to 1} \frac{1 + 2 x - x - 2 x^{2}}{(1 - x)} = \lim_{x \to 1} \frac{1 (1 + 2 x) - x (1 + 2 x)}{(1 - x)} = \lim_{x \to 1} \frac{(1 + 2 x) (1 - x)}{(1 - x)} = \lim_{x \to 1} (1 + 2 x)$

Step 3. Let us use the sum rule in the obtained expression.

$\lim_{x \to 1} (1 + 2 x) = \lim_{x \to 1} (1) + \lim_{x \to 1} (2 x) = 1 + 2 (1) = 1 + 2 = 3$

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Calculate each of the limits in Exercises $29 - 70$ .
$\lim_{x \to 1} \frac{x + x^{2} - 2 x^{3}}{x - x^{2}}$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. The limit is calculated as below:

Step 3. Let us use the sum rule in the obtained expression.

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Calculate each of the limits in Exercises 29-70.limx→1x+x2-2x3x-x2.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The limit is calculated as below:

Step 3. Let us use the sum rule in the obtained expression.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

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Study anywhere. Anytime. Across all devices.

Calculate each of the limits in Exercises $29 - 70$ .
$\lim_{x \to 1} \frac{x + x^{2} - 2 x^{3}}{x - x^{2}}$ .