Math Problem Statement
Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it. lim x→0+ (cos(x))3/x2
Solution
To find the limit
we first substitute :
which indicates that the limit approaches infinity. However, to confirm whether the limit diverges or has a specific behavior, we can analyze the function further.
As approaches , approaches . Therefore, approaches as well. The denominator approaches . Thus, we have:
To use l'Hôpital's Rule, we can re-examine the limit. Since both the numerator and denominator approach as , we can rewrite it as follows:
- Apply l'Hôpital's Rule:
We differentiate the numerator and denominator:
Thus, we have:
Now substituting :
The numerator approaches (since ) and the denominator approaches as well, allowing us to apply l'Hôpital's Rule again.
- Apply l'Hôpital's Rule a second time:
Differentiate again:
- The numerator becomes:
- The denominator becomes:
Now the limit is:
As , this evaluates to:
However, since we want the limit of the original function as , we find:
Thus, the final answer is:
Summary:
The limit diverges to positive infinity.
Do you want further details on this solution or have any questions? Here are some related questions you might consider:
- What is the behavior of as approaches other values?
- How does l'Hôpital's Rule apply to different types of indeterminate forms?
- Can you explain the Taylor series expansion for ?
- What are other limits involving trigonometric functions?
- How does the limit change if we consider from both sides?
Tip: Always check the form of the limit before deciding on the method to apply!
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometric Functions
L'Hôpital's Rule
Formulas
lim (cos(x))^3/x^2 as x→0+
L'Hôpital's Rule: lim f(x)/g(x) = lim f'(x)/g'(x) if indeterminate
Theorems
L'Hôpital's Rule
Suitable Grade Level
Grades 11-12