Math Problem Statement
lim(x -> 0+) (cosx - 2x)/(2xsinx+x^2cosx)
Solution
To evaluate the limit:
we will proceed step-by-step.
Step 1: Substitution at
Direct substitution of gives: This is indeterminate since the denominator approaches . We proceed with L’Hôpital’s Rule as both the numerator and denominator tend to as .
Step 2: Apply L’Hôpital’s Rule
Differentiate the numerator and the denominator with respect to :
Numerator:
Denominator:
Simplify the derivative of the denominator:
Step 3: Rewrite the limit
The limit becomes:
Step 4: Substitution at again
Substituting :
- Numerator: ,
- Denominator: .
Again, this is indeterminate, so we apply L’Hôpital’s Rule once more.
Step 5: Apply L’Hôpital’s Rule again
Differentiate the numerator and denominator again:
Numerator:
Denominator:
Differentiate : Simplify:
Step 6: Rewrite the limit
The limit becomes:
Step 7: Substitution at
Substitute :
- Numerator: ,
- Denominator: .
Thus, the limit is:
Final Answer:
Let me know if you would like further clarification or additional examples. 😊
Five Related Questions:
- How is L’Hôpital’s Rule applied to solve indeterminate forms?
- What are common indeterminate forms encountered in calculus?
- Can you evaluate ?
- How do trigonometric approximations (e.g., for small ) help in solving limits?
- What are alternative methods to solve limits when L’Hôpital’s Rule fails?
Tip:
When using L’Hôpital’s Rule, always check that the function is in an indeterminate form like or before applying the rule.
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Math Problem Analysis
Mathematical Concepts
Limits
L'Hopital's Rule
Trigonometric Functions
Indeterminate Forms
Formulas
L'Hopital's Rule
cos(x)
sin(x)
Theorems
L'Hopital's Rule
Suitable Grade Level
Grades 11-12
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