Math Problem Statement
Solve this
Solution
To solve the given limit, we need to analyze the behavior of the function as approaches infinity:
Step-by-Step Solution:
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Simplify the Numerator and Denominator:
First, simplify the numerator: [ \sqrt[3]{2x^2 - x^3} = \sqrt[3]{-x^3 (1 - \frac{2x^2}{x^3})} = \sqrt[3]{-x^3 (1 - \frac{2}{x})} ] As , approaches 0. Thus: [ \sqrt[3]{2x^2 - x^3} \approx \sqrt[3]{-x^3} = -x ]
Next, simplify the denominator:
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Evaluate the Dominant Terms:
For large , the dominant terms in the numerator and denominator will dictate the behavior of the fraction:
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Take the Limit:
As :
Conclusion:
Would you like any additional details or have any questions about this solution?
Here are 5 questions you might ask next:
- How do we handle limits involving square roots or cube roots?
- What are the general steps for evaluating limits at infinity?
- How can L'Hôpital's rule be applied to more complex limits?
- What are some common pitfalls when simplifying algebraic expressions?
- How do we determine the dominant term in a polynomial as ?
Tip: When simplifying limits, always identify the dominant terms that grow the fastest as . This can often simplify the process significantly.
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Math Problem Analysis
Mathematical Concepts
Limits
Infinity
Cube Roots
Polynomials
Formulas
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Theorems
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Suitable Grade Level
Advanced High School
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