Math Problem Statement
Evaluate the following limit. Use l'Hôpital's Rule when it is convenient and applicable. ModifyingBelow lim With x right arrow 1limx→1 StartFraction ln x Over 14 x minus x squared minus 13 EndFraction lnx 14x−x2−13 Question content area bottom Part 1 Use l'Hôpital's Rule to rewrite the given limit so that it is not an indeterminate form. ModifyingBelow lim With x right arrow 1limx→1 StartFraction ln x Over 14 x minus x squared minus 13 EndFraction lnx 14x−x2−13equals=ModifyingBelow lim With x right arrow 1limx→1left parenthesis nothing right parenthesisenter your response here
Solution
To evaluate the limit
we first substitute :
This gives us the indeterminate form . Thus, we can apply l'Hôpital's Rule.
Step 1: Differentiate the numerator and the denominator
- Numerator: The derivative of is .
- Denominator: The derivative of is .
Step 2: Rewrite the limit using l'Hôpital's Rule
Using l'Hôpital's Rule, we rewrite the limit:
Step 3: Evaluate the new limit
Now we substitute :
Conclusion
Thus, the limit evaluates to
If you have any questions or need further details, feel free to ask!
Related Questions:
- How do you apply l'Hôpital's Rule in different types of indeterminate forms?
- What are the conditions under which l'Hôpital's Rule can be used?
- Can you provide examples of limits that do not require l'Hôpital's Rule?
- What is the significance of continuity in evaluating limits?
- How does the derivative of a function relate to the concept of limits?
Tip:
Always check if you can simplify the expression before applying l'Hôpital's Rule, as it may lead to an easier evaluation.
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Math Problem Analysis
Mathematical Concepts
Calculus
Limits
Indeterminate Forms
Formulas
l'Hôpital's Rule
Limit definition
Theorems
l'Hôpital's Rule
Suitable Grade Level
Grades 11-12