Math Problem Statement
Solve this
Solution
The given limit problem is:
Step 1: Substituting
First, check the form of the limit by directly substituting :
- ,
- .
Substituting into the expression gives:
This is an indeterminate form , so we can apply L'Hôpital's Rule.
Step 2: Applying L'Hôpital's Rule
L'Hôpital's Rule states that if a limit results in an indeterminate form , we can differentiate the numerator and the denominator with respect to , and then re-evaluate the limit.
The expression is:
Differentiate the numerator:
The derivative of with respect to is:
Differentiate the denominator:
The derivative of with respect to is:
So, applying L'Hôpital's Rule gives:
Simplifying this:
Step 3: Evaluating the Limit
Now substitute into the expression. Since , we get:
Again, this results in an indeterminate form, so we apply L'Hôpital's Rule a second time.
Step 4: Second Application of L'Hôpital's Rule
Differentiate the numerator and denominator again.
Numerator:
The derivative of is:
Denominator:
The derivative of is:
So the new limit is:
Now substitute into this expression:
Final Answer:
Let me know if you have any questions or need more details.
Here are 5 related questions to expand your understanding:
- How does L'Hôpital's Rule help in solving indeterminate forms?
- What other techniques can be used to solve limits besides L'Hôpital's Rule?
- Can the limit be evaluated using series expansions around ?
- What are the conditions under which L'Hôpital's Rule can be applied?
- What would happen if we tried to solve this limit using a Taylor series expansion?
Tip: Always check if a limit is indeterminate before applying L'Hôpital’s Rule—other methods like factorization or series expansion might be simpler!
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometry
Calculus
L'Hopital's Rule
Formulas
L'Hopital's Rule
Derivative of sin x
Derivative of a power function
Theorems
L'Hopital's Rule
Suitable Grade Level
Grades 11-12 or introductory college calculus
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