Math Problem Statement
Use the epsilon definition to find and prove lim n to infinity ((n^2+(square root n))/(n^2 +1))
Solution
To prove
using the epsilon () definition of limits, we need to show that for any , there exists a positive integer such that for all ,
Step 1: Simplify the Expression
Let's start by simplifying the expression inside the absolute value:
So, we need to make the expression smaller than .
Step 2: Analyze the Simplified Expression as
As becomes very large:
- The denominator grows approximately like .
- The numerator grows much more slowly, since .
Thus, as , , suggesting that we can find an large enough for the expression to be less than any positive .
Step 3: Make the Inequality Precise
To satisfy , we can bound the expression as follows:
Now simplify this expression further:
Step 4: Choose to Satisfy the Epsilon Condition
We want . This inequality is satisfied if is large enough, specifically if:
Let . Then for any , we have:
Conclusion
By the epsilon definition of a limit, we have shown that for any , there exists an such that for all ,
Thus,
Would you like more details on any specific part of this proof?
Here are five questions for further exploration:
- How can we apply the epsilon-delta definition for limits involving functions of rather than sequences?
- What would change if the numerator had a higher power term, like instead of ?
- How does the epsilon proof change for a limit approaching a non-infinity value?
- What are other ways to analyze limits involving square roots?
- How does this approach apply to limits in calculus of continuous functions?
Tip: Always examine both numerator and denominator growth rates to simplify limit calculations involving sequences.
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Math Problem Analysis
Mathematical Concepts
Limit of Sequences
Epsilon-Delta Definition of Limits
Asymptotic Behavior
Square Roots
Formulas
lim_{n->∞} f(n) = L
|f(n) - L| < ε
Theorems
Epsilon-Delta Definition of Limits
Suitable Grade Level
Grades 11-12
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