Math Problem Statement
Find the value
Solution
We are given the limit:
To evaluate this limit, we can use standard limit techniques. One common approach for this type of problem is applying a known trigonometric limit formula or L'Hopital's Rule, since direct substitution gives an indeterminate form .
Method 1: Using Trigonometric Identity
First, we use the trigonometric identity:
Thus, the given limit becomes:
We now divide both the numerator and denominator by :
Now using the standard limit:
\lim_{{x \to 0}} \frac{\sin{y}}{y} = 1 \quad \text{(where y = \frac{x}{2})},
we get:
Therefore, the value of the limit is 0.
Method 2: Using L'Hopital's Rule
Since the given limit initially results in the indeterminate form , we can apply L'Hopital's Rule, which involves taking the derivative of the numerator and the denominator.
- Derivative of the numerator is .
- Derivative of the denominator is .
Thus, applying L'Hopital's Rule, the limit becomes:
So, the value of the limit is 0.
Let me know if you want more details or further clarification!
Here are 5 relative questions that can expand your understanding:
- What is L'Hopital's Rule and when should it be applied in limits?
- How can we use Taylor series to expand functions like and evaluate limits?
- What are some other indeterminate forms besides ?
- Why is the limit considered fundamental in calculus?
- How does the small-angle approximation near zero help in solving limits?
Tip: Memorizing common trigonometric limits and identities can greatly speed up solving problems involving limits like this!
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometry
L'Hopital's Rule
Formulas
lim(x -> 0) (sin(x) / x) = 1
1 - cos(x) = 2sin^2(x/2)
Theorems
L'Hopital's Rule
Suitable Grade Level
Grade 11-12
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