Chapter 2: Q. 1TF (page 223)

Show that each of the following limits is of the form $\frac{0}{0}$ and then use L’Hopital’s rule to calculate the limit: ˆ
$\lim_{x \to 0} \frac{x^{3}}{1 - 2^{x}}$

Short Answer

Expert verified

Hence the solution is $\frac{6}{- \log (2) \log (2) \log (2)}$

Step by step solution

Step 1. Given information

The given information is $\lim_{x \to 0} \frac{x^{3}}{1 - 2^{x}}$

Step 2. Calculate the limit

$\lim_{x \to 0} \frac{x^{3}}{1 - 2^{x}} \lim_{x \to 0} \frac{\frac{d (x^{3})}{d x}}{\frac{d (1 - 2^{x})}{d x}} = \lim_{x \to 0} \frac{3 x^{2}}{- 2^{x} \log (2)} \lim_{x \to 0} \frac{\frac{d (3 x^{2})}{d x}}{\frac{d (- 2^{x} \log (2))}{d x}} = \lim_{x \to 0} \frac{6 x}{- 2^{x} \log (2) \log (2)} \lim_{x \to 0} \frac{\frac{d (6 x)}{d x}}{\frac{d (- 2^{x} \log (2) \log (2))}{d x}} = \lim_{x \to 0} \frac{6}{- 2^{x} \log (2) \log (2) \log (2)} = \frac{6}{- \log (2) \log (2) \log (2)}$

Step 3. The solution

The solution is $\frac{6}{- \log (2) \log (2) \log (2)}$

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Show that each of the following limits is of the form $\frac{0}{0}$ and then use L’Hopital’s rule to calculate the limit: ˆ
$\lim_{x \to 0} \frac{x^{3}}{1 - 2^{x}}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculate the limit

Step 3. The solution

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Show that each of the following limits is of the form 00and then use L’Hopital’s rule to calculate the limit: ˆlimx→0x31-2x

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculate the limit

Step 3. The solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Show that each of the following limits is of the form $\frac{0}{0}$ and then use L’Hopital’s rule to calculate the limit: ˆ
$\lim_{x \to 0} \frac{x^{3}}{1 - 2^{x}}$