Q. 20 Calculate each of the limits in ... [FREE SOLUTION]

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Chapter 3: Q. 20 (page 310)

Calculate each of the limits in Exercises 15鈥�20
(a) using L鈥橦opital鈥檚 rule and (b) without using L鈥橦opital鈥檚 rule.
$\lim_{x 鈫� 鈭�} \frac{2^{x} - 1}{4^{x} - 1}$

Short Answer

Expert verified

Part (a) The value of $\lim_{x 鈫� 鈭�} \frac{2^{x} - 1}{4^{x} - 1}$ using L'Hopitals rule $= \frac{1}{2}$

Part (b) The value of $\lim_{x 鈫� 鈭�} \frac{2^{x} - 1}{4^{x} - 1}$ without using L'Hopitals rule $= \frac{1}{2}$

Step by step solution

Part (a) Step 1. Given information

The given limit $\lim_{x 鈫� 鈭�} \frac{2^{x} - 1}{4^{x} - 1}$

Part (a) Step 2. Finding the value of the given limit using L’Hopital’s rule

The limit using L鈥橦opitals rule is given below:

$\lim_{x 鈫� 0} \frac{2^{x} - 1}{4^{x} - 1} = \lim_{x 鈫� 0} \frac{2^{x} \ln 2}{4^{x} \ln 4} = \frac{\ln 2}{\ln 4} \lim_{x 鈫� 0} \frac{2^{x}}{4^{x}} = \frac{\ln 2}{\ln 4} \lim_{x 鈫� 0} \frac{2^{x}}{2^{2 x}} = \frac{\ln 2}{\ln 4} \lim_{x 鈫� 0} 2^{x - 2 x} = \frac{\ln 2}{\ln 4} \lim_{x 鈫� 0} 2^{- x} = \frac{\ln 2}{\ln 2^{2}} . 1 = \frac{\ln 2}{2 \ln 2} = \frac{1}{2}$

Therefore,the value of limit $\frac{1}{2}$

Part (b) Step 1. Given information

The given limit $\lim_{x 鈫� 鈭�} \frac{2^{x} - 1}{4^{x} - 1}$

Part (b) Step 2. Finding the value of the given limit without using L’Hopital’s rule

The limit without using L'Hopitals rule is given below

$\lim_{x 鈫� 0} \frac{2^{x} - 1}{4^{x} - 1} = \lim_{x 鈫� 0} \frac{2^{x} - 1}{{(2^{x})}^{2} - 1^{2}} = \lim_{x 鈫� 0} \frac{2^{x} - 1}{(2^{x} + 1) (2^{x} - 1)} = \lim_{x 鈫� 0} \frac{1}{2^{x} + 1} = \frac{1}{1 + 1} = \frac{1}{2}$

Therefore,the value of the limit $= \frac{1}{2}$

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91影视

Calculate each of the limits in Exercises 15鈥�20
(a) using L鈥橦opital鈥檚 rule and (b) without using L鈥橦opital鈥檚 rule.
$\lim_{x 鈫� 鈭�} \frac{2^{x} - 1}{4^{x} - 1}$

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) Step 2. Finding the value of the given limit using L’Hopital’s rule

Part (b) Step 1. Given information

Part (b) Step 2. Finding the value of the given limit without using L’Hopital’s rule

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91影视

Calculate each of the limits in Exercises 15鈥�20(a) using L鈥橦opital鈥檚 rule and (b) without using L鈥橦opital鈥檚 rule.limx鈫�鈭�2x-14x-1

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) Step 2. Finding the value of the given limit using L’Hopital’s rule

Part (b) Step 1. Given information

Part (b) Step 2. Finding the value of the given limit without using L’Hopital’s rule

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

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Probability and Statistics

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Calculate each of the limits in Exercises 15鈥�20
(a) using L鈥橦opital鈥檚 rule and (b) without using L鈥橦opital鈥檚 rule.
$\lim_{x 鈫� 鈭�} \frac{2^{x} - 1}{4^{x} - 1}$