Chapter 3: Q. 26 (page 314)

L’Hˆopital’s Rule limit calculations: Calculate each of the
limits that follow. Some of these limits are easier to calculate
by using L’Hopital’s rule, and some are not.
$\lim_{x \to 0} \frac{3^{x} - 1}{2^{x} - 1}$

Short Answer

Expert verified

Limit is $\frac{\ln 3}{\ln 2}$ and it is easier to find using L'Hospitals rule.

Step by step solution

Step 1. Given information

An expression is given as $\lim_{x \to 0} \frac{3^{x} - 1}{2^{x} - 1}$

Step 2. Evaluating limit

Use properties of limit and simplify as,

$\lim_{x \to 0} \frac{3^{x} - 1}{2^{x} - 1} = \lim_{x \to 0} \frac{3^{x} - 1}{2^{x} - 1} = \lim_{x \to 0} \frac{3^{x} \ln 3 - 0}{2^{x} \ln 2 - 0} = \frac{\ln 3}{\ln 2} \lim_{x \to 0} (\frac{3^{x}}{2^{x}}) = \frac{\ln 3}{\ln 2} \times \frac{3^{0}}{2^{0}} = \frac{\ln 3}{\ln 2}$

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L’Hˆopital’s Rule limit calculations: Calculate each of the
limits that follow. Some of these limits are easier to calculate
by using L’Hopital’s rule, and some are not.
$\lim_{x \to 0} \frac{3^{x} - 1}{2^{x} - 1}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating limit

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L’Hˆopital’s Rule limit calculations: Calculate each of thelimits that follow. Some of these limits are easier to calculateby using L’Hopital’s rule, and some are not.limx→03x-12x-1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating limit

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Most popular questions from this chapter

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L’Hˆopital’s Rule limit calculations: Calculate each of the
limits that follow. Some of these limits are easier to calculate
by using L’Hopital’s rule, and some are not.
$\lim_{x \to 0} \frac{3^{x} - 1}{2^{x} - 1}$