Math Problem Statement
Solution
We are asked to find the limit:
Step-by-Step Solution:
-
Apply Direct Substitution:
Substituting into the expression, we get: This results in an indeterminate form , so we need to apply L'Hôpital's Rule. -
L'Hôpital's Rule:
L'Hôpital's Rule states that if a limit results in an indeterminate form , we can take the derivative of the numerator and the denominator separately and then find the limit of that expression. Thus, we compute the derivatives of and .- Derivative of is .
- Derivative of is (using the chain rule).
Now, apply L'Hôpital's Rule:
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Evaluate the New Limit:
Substituting into this new expression:
Final Answer:
Would you like more details or have any further questions?
Here are 5 related questions to consider:
- How do you apply L'Hôpital's Rule to more complex indeterminate forms?
- What are the conditions for using L'Hôpital's Rule?
- Can you find limits involving different trigonometric functions?
- How would you solve this limit without L'Hôpital's Rule, using series expansions?
- How does the limit change if the denominator was for any constant ?
Tip: Always check if the limit is indeterminate before applying L'Hôpital's Rule.
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometric Functions
Indeterminate Forms
Formulas
L'Hôpital's Rule: If a limit results in 0/0 or ∞/∞, take the derivative of the numerator and denominator separately
Derivative of sin(x) = cos(x)
Derivative of sin(πx) = πcos(πx)
Theorems
L'Hôpital's Rule
Suitable Grade Level
Grades 11-12
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