Math Problem Statement
Evaluate the limit as x approaches 0 for the expression (x^2 * sin(x) * (1/2) * sin(x) * sqrt(x)) / x^3.
Solution
The given limit expression is:
Let's simplify the expression step by step:
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The expression in the numerator is: This can be rewritten as: Which simplifies to:
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The full expression for the limit becomes:
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Simplify the fraction:
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Now, as , we know that: So, using this fact, we can approximate near zero:
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Therefore, the limit becomes:
Thus, the value of the limit is:
Would you like more details on any step? Here are some related questions:
- How do we apply limits involving trigonometric functions like ?
- Why does hold?
- What are other methods to simplify limits involving ?
- How does L'Hopital's Rule apply to this kind of limit problem?
- Could Taylor series expansions help simplify this expression?
Tip: When working with limits involving trigonometric functions, knowing common limits like can save time!
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometry
Formulas
lim (sin(x)/x) as x -> 0 = 1
Theorems
Small-angle approximation for sine
Suitable Grade Level
University level (Calculus)