Math Problem Statement
Lim x->0 cosx-cos2x/x²
Solution
We are tasked with evaluating the limit:
Step-by-step solution:
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Apply L'Hopital's Rule: Since both the numerator and the denominator approach 0 as (an indeterminate form of ), we can apply L'Hopital's Rule. This requires taking the derivative of both the numerator and the denominator.
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First derivative of the numerator: The numerator is .
- The derivative of is .
- The derivative of is (using the chain rule).
So, the derivative of the numerator is:
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First derivative of the denominator: The denominator is , and the derivative of is .
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New limit expression: After applying L'Hopital's Rule once, we now evaluate the following limit: This is still an indeterminate form , so we apply L'Hopital's Rule again.
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Second derivative of the numerator: The numerator is now .
- The derivative of is .
- The derivative of is (using the chain rule again).
So, the second derivative of the numerator is:
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Second derivative of the denominator: The second derivative of is just 2.
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Final limit expression: After applying L'Hopital's Rule again, we now evaluate: At :
Substituting these values into the limit:
Conclusion:
The value of the limit is:
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometric Functions
L'Hopital's Rule
Derivatives
Formulas
L'Hopital's Rule: \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}
Derivative of cos(x): -sin(x)
Chain rule for trigonometric functions
Theorems
L'Hopital's Rule
Chain Rule
Suitable Grade Level
Grades 11-12 and early university level
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