Evaluate the following limit. Use l'Hôpital's Rule when it is convenient and applicable. 2x - 4x - lim x4 14x-8 X→-1X" +2x – 4x- - 10x –5 How should the given limit be evaluated? Select the correct choice below and, if necessary, fill in the answer box to complete your ch O A Multiply the expression by a unit fraction to obtain lim X→ - 1 O B. Use direct substitution. U C. Use l'Hôpital's Rule more than once to rewrite the limit in its final form as lim ) X -1 O D. Use l'Hôpital's Rule exactly once to rewrite the limit as lim X→ - 1 Evaluate the limit. 2x - 4x2 - 14x-8 lim x* +2x - 4x2 -10x-5 (Type an exact answer.) 2
Evaluate the following limit. Use l'Hôpital's Rule when it is convenient and applicable. 2x - 4x - lim x4 14x-8 X→-1X" +2x – 4x- - 10x –5 How should the given limit be evaluated? Select the correct choice below and, if necessary, fill in the answer box to complete your ch O A Multiply the expression by a unit fraction to obtain lim X→ - 1 O B. Use direct substitution. U C. Use l'Hôpital's Rule more than once to rewrite the limit in its final form as lim ) X -1 O D. Use l'Hôpital's Rule exactly once to rewrite the limit as lim X→ - 1 Evaluate the limit. 2x - 4x2 - 14x-8 lim x* +2x - 4x2 -10x-5 (Type an exact answer.) 2
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![### Evaluating Limits Using L'Hôpital's Rule
**Problem Statement:**
Evaluate the following limit. Use L'Hôpital's Rule when it is convenient and applicable.
\[ \lim_{x \to 1} \frac{2x^3 - 4x^2 - 14x - 8}{x^4 + 2x^3 - 4x^2 - 10x - 5} \]
**Question:**
How should the given limit be evaluated? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
1. [ ] Multiply the expression by a unit fraction to obtain \( \lim_{x \to 1} \left( \right) \)
2. [ ] Use direct substitution.
3. [ ] Use L'Hôpital's Rule more than once to rewrite the limit in its final form as \( \lim_{x \to 1} \left( \right) \)
4. [ ] Use L'Hôpital's Rule exactly once to rewrite the limit as \( \lim_{x \to 1} \left( \right) \)
**Final Calculation:**
Evaluate the limit:
\[ \lim_{x \to 1} \frac{2x^3 - 4x^2 - 14x - 8}{x^4 + 2x^3 - 4x^2 - 10x - 5} = \boxed{\frac{5}{2}} \]
**Explanation:**
To evaluate the limit using L'Hôpital's Rule, we first confirm that direct substitution results in an indeterminate form (such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \)). If it does, we apply L'Hôpital's Rule, which states that:
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}, \]
provided the limit on the right-hand side exists.
Depending on the complexity of the function, it may be necessary to apply L'Hôpital's Rule more than once. Therefore, option 3 suggests evaluating the limit by using L'Hôpital's Rule multiple times until the expression is simplified enough to compute the limit.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb02fb262-35a8-4c2d-87d7-48cdf0e8c8b9%2F2a085741-589f-4c55-affd-072f1deb1dbf%2Fo30hwh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Evaluating Limits Using L'Hôpital's Rule
**Problem Statement:**
Evaluate the following limit. Use L'Hôpital's Rule when it is convenient and applicable.
\[ \lim_{x \to 1} \frac{2x^3 - 4x^2 - 14x - 8}{x^4 + 2x^3 - 4x^2 - 10x - 5} \]
**Question:**
How should the given limit be evaluated? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
1. [ ] Multiply the expression by a unit fraction to obtain \( \lim_{x \to 1} \left( \right) \)
2. [ ] Use direct substitution.
3. [ ] Use L'Hôpital's Rule more than once to rewrite the limit in its final form as \( \lim_{x \to 1} \left( \right) \)
4. [ ] Use L'Hôpital's Rule exactly once to rewrite the limit as \( \lim_{x \to 1} \left( \right) \)
**Final Calculation:**
Evaluate the limit:
\[ \lim_{x \to 1} \frac{2x^3 - 4x^2 - 14x - 8}{x^4 + 2x^3 - 4x^2 - 10x - 5} = \boxed{\frac{5}{2}} \]
**Explanation:**
To evaluate the limit using L'Hôpital's Rule, we first confirm that direct substitution results in an indeterminate form (such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \)). If it does, we apply L'Hôpital's Rule, which states that:
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}, \]
provided the limit on the right-hand side exists.
Depending on the complexity of the function, it may be necessary to apply L'Hôpital's Rule more than once. Therefore, option 3 suggests evaluating the limit by using L'Hôpital's Rule multiple times until the expression is simplified enough to compute the limit.
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