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Chapter 2: Q. 1TF (page 223)

Show that each of the following limits is of the form $\frac{0}{0}$ and then use L鈥橦opital鈥檚 rule to calculate the limit: 藛
$\lim_{x 鈫� 0} \frac{x^{3}}{1 - 2^{x}}$

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information

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

02

Step 2. Calculate the limit

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

03

Step 3. The solution

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

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Most popular questions from this chapter

Use the definition of the derivative to find $f'$ for each function f in Exercises 39-54

$f (x) = - x^{3} - x + 1$

Think about what you did today and how far north you were from your house or dorm throughout the day. Sketch a graph that represents your distance north from your house or dorm over the course of the day, and explain how the graph reflects what you did today. Then sketch a graph of your velocity.

Prove, in two ways, that the power rule holds for negative integer powers

a) by using the $z 鈫� x$ definition of the derivative

b) by using the $h 鈫� 0$ definition of the derivative

For each function $f$ graphed in Exercises 65-68, determine the values of $x$ at which $f$ fails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = \frac{1}{\sqrt{x^{2} + 1}}$

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