Math Problem Statement
lim x -> 0 (3 - 3cos^2 x)/(x ^ 2) adalah...
Solution
We are tasked with finding the limit:
Step 1: Factor out the constant
We can factor out the constant from the numerator:
Step 2: Use a trigonometric identity
Recall that (from the Pythagorean identity). Substituting this into the expression:
Step 3: Simplify the limit
We know that , so we can rewrite the expression:
Thus, the value of the limit is:
Would you like further clarification on any step?
Here are five related questions you can explore:
- What is the limit of ?
- How can we derive using L'Hopital's Rule?
- What are some common trigonometric limits and how are they derived?
- How does L'Hopital's Rule apply to limits involving indeterminate forms?
- What is the Maclaurin series for and , and how can it be used in limits?
Tip: For small angles, trigonometric functions like and can often be approximated using Taylor or Maclaurin series expansions.
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometric Identities
L'Hopital's Rule
Formulas
lim_{x \to 0} \frac{\sin x}{x} = 1
1 - \cos^2 x = \sin^2 x
Theorems
Pythagorean Identity
Trigonometric Limits
Suitable Grade Level
Grades 11-12
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