Math Problem Statement
Solution
Let's evaluate the three limits one by one. Here are the step-by-step solutions:
1. Limit 1:
- For , the denominator remains positive because it is squared, but the numerator depends on .
- When , becomes slightly negative, and remains positive.
- Therefore, (slightly less than 4 from the left).
So,
Conclusion:
2. Limit 2:
- The dominant term in both the numerator and denominator is as .
- Divide both numerator and denominator by to simplify:
- As , terms involving , so the expression simplifies to:
Conclusion:
3. Limit 3:
- For , the dominant term in the numerator is , and in the denominator, dominates.
- Divide both numerator and denominator by :
Simplify each term:
- as .
- as .
Thus, the limit simplifies to:
Conclusion:
Final Answers:
Do you need further details or have any additional questions?
Here are 5 related questions for practice:
- Evaluate .
- Find .
- Calculate .
- Evaluate .
- Find .
Tip:
When evaluating limits involving large , always focus on the dominant terms (highest powers of ) in the numerator and denominator.
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Math Problem Analysis
Mathematical Concepts
Limits
Dominant terms in polynomials
Behavior of functions as x approaches zero or infinity
Formulas
For large x, dominant terms determine the behavior of polynomials.
Limits at infinity: Divide numerator and denominator by the highest power of x.
For x approaching 0, consider sign and behavior of terms.
Theorems
Limits of rational functions
Behavior of polynomials at infinity
Suitable Grade Level
Undergraduate Calculus or AP Calculus (Grade 11-12)
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