Math Problem Statement
Solution
To solve the given limit problem:
Step-by-step process:
-
Examine the degrees of the polynomial:
- In the numerator: has a degree of 1.
- In the denominator: has a degree of 3.
-
Leading terms for large :
- The leading term in the numerator is .
- The leading term in the denominator is .
-
Simplify by dividing numerator and denominator by (the highest power of in the denominator):
-
Simplify each term:
- Numerator: .
- Denominator: .
-
Take the limit as :
- The terms involving , , and in both the numerator and denominator will approach 0 as .
- Thus, the numerator approaches 0.
- The denominator approaches .
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Conclusion: The limit is:
Therefore, the correct answer is 0.
Would you like more details on any step, or do you have any questions?
Related Questions:
- How do you determine which term dominates in the limit at infinity?
- What would happen if the degrees of the numerator and denominator were the same?
- How do you handle limits approaching infinity versus negative infinity?
- What happens when the numerator has a higher degree than the denominator?
- Can we use L'Hopital's rule to solve this limit?
Tip: When solving limits involving polynomials, always look for the highest degree terms, as they dominate the behavior at large values of .
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Math Problem Analysis
Mathematical Concepts
Limits at Infinity
Polynomials
Formulas
Leading term comparison of polynomials
Theorems
Limit of polynomials as x approaches infinity
Suitable Grade Level
Grades 11-12 or College Level
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