Compute the following limits using I'H\^opital's rule if appropriate. Use INF to denote co and MINF to denote-co. 1- cos(2x) lim x-0 1- cos(8x) lim x-1 9*8*1 x² - 1 =
Compute the following limits using I'H\^opital's rule if appropriate. Use INF to denote co and MINF to denote-co. 1- cos(2x) lim x-0 1- cos(8x) lim x-1 9*8*1 x² - 1 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Calculating Limits Using L'Hôpital's Rule
#### Task:
Compute the following limits using L'Hôpital's rule if appropriate. Use **INF** to denote \( \infty \) and **MINF** to denote \( -\infty \).
1. \[
\lim_{{x \to 0}} \frac{1 - \cos(2x)}{1 - \cos(8x)} = \, \boxed{}
\]
2. \[
\lim_{{x \to 1}} \frac{9^x - 8^x - 1}{x^2 - 1} = \, \boxed{}
\]
#### Explanation:
The above expressions require you to find the limits as \( x \) approaches a specified value. L'Hôpital's rule may be used if you encounter an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
- **L'Hôpital's Rule** can be applied when both the numerator and denominator approach 0 or \( \infty \).
- Differentiate the numerator and the denominator separately until the limit can be evaluated.
#### Notes:
- The boxed areas are for you to fill in after solving the limits.
- Make sure to verify the conditions for applying L'Hôpital's rule.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58cffe43-d701-4c21-9740-08fe8d98ee79%2Fd495a3a6-082d-4d11-b0f9-6481af3e1e4d%2Fgsa0sxl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating Limits Using L'Hôpital's Rule
#### Task:
Compute the following limits using L'Hôpital's rule if appropriate. Use **INF** to denote \( \infty \) and **MINF** to denote \( -\infty \).
1. \[
\lim_{{x \to 0}} \frac{1 - \cos(2x)}{1 - \cos(8x)} = \, \boxed{}
\]
2. \[
\lim_{{x \to 1}} \frac{9^x - 8^x - 1}{x^2 - 1} = \, \boxed{}
\]
#### Explanation:
The above expressions require you to find the limits as \( x \) approaches a specified value. L'Hôpital's rule may be used if you encounter an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
- **L'Hôpital's Rule** can be applied when both the numerator and denominator approach 0 or \( \infty \).
- Differentiate the numerator and the denominator separately until the limit can be evaluated.
#### Notes:
- The boxed areas are for you to fill in after solving the limits.
- Make sure to verify the conditions for applying L'Hôpital's rule.
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