Math Problem Statement
Solution
Let's solve the two limit problems from the image.
Problem 8:
We are asked to find:
First, let's simplify the expression.
- using the identity .
- As , , so the numerator tends to 0.
- In the denominator, as :
- .
- Hence, .
Both the numerator and denominator tend to 0, so we can apply L'Hopital's Rule.
Now, differentiate the numerator and denominator with respect to :
- The derivative of the numerator, , is , and .
- The derivative of the denominator, , is:
Thus, applying L'Hopital's Rule, we get: Now, substitute into this expression:
- ,
- .
Hence, the limit becomes:
So, the answer is C. .
Problem 9:
We are asked to find:
First, substitute :
- The numerator becomes .
- The denominator becomes .
Since both the numerator and denominator tend to 0, we can apply L'Hopital's Rule.
Differentiate the numerator and denominator:
-
The derivative of using the product rule is: At , and , so this becomes:
-
The derivative of the denominator, , is: At , this becomes .
Now, applying L'Hopital's Rule, we get:
So, the answer is C. .
Let me know if you want further details or clarifications. Here are 5 related questions you might find interesting:
- How do you determine when to apply L'Hopital's Rule?
- What is the geometric interpretation of limits involving trigonometric functions?
- How can you apply Taylor series to simplify limit problems?
- What are the conditions for applying L'Hopital’s Rule?
- Can you solve limits using factorization instead of L'Hopital's Rule?
Tip: Always check if direct substitution works before using advanced techniques like L'Hopital’s Rule.
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometric Limits
L'Hopital's Rule
Differentiation
Formulas
L'Hopital's Rule: lim x → a [f(x)/g(x)] = lim x → a [f'(x)/g'(x)] if f(a)=g(a)=0
Derivative of sin(x) and cos(x)
Derivative of tan(x)
Theorems
L'Hopital's Rule
Trigonometric Limit Properties
Suitable Grade Level
Grade 12/University
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