Math Problem Statement
Solution
The limit expression provided in the image is:
Step-by-Step Solution:
We are tasked with finding:
Step 1: Rewrite the expression
Let us rewrite the base of the exponent in a more convenient form:
As , the term grows large, making also large. However, the exponent tends to zero, so this is an indeterminate form of type .
Step 2: Apply logarithms
To resolve the indeterminate form, we can apply logarithms. Let . Taking the natural logarithm of both sides:
Step 3: Simplify the logarithm
For large , we can approximate as approximately . Hence:
Thus, the expression becomes:
Since is a constant, the limit will depend on the behavior of .
Step 4: Evaluate the limit
We now need to evaluate:
This is a well-known limit, and it evaluates to 0, since grows slower than :
Thus, we have:
Step 5: Final result
Exponentiating both sides gives:
Therefore, the original limit evaluates to:
Conclusion:
The value of the limit is .
Would you like further details on this solution, or do you have any specific questions? Here are some follow-up questions you might find useful:
- What is the significance of the logarithmic approach in limit problems?
- Why is the form indeterminate, and how do we resolve such cases?
- How does L'Hôpital's Rule relate to evaluating limits like this?
- What are other common techniques to handle limits at infinity?
- How do logarithmic approximations help in simplifying limit problems?
Tip: Whenever you encounter an indeterminate form like , , or , applying logarithms can often simplify the expression and make it easier to compute the limit.
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Math Problem Analysis
Mathematical Concepts
Limits
Exponential Growth
Logarithms
Formulas
lim (1 + x/2)^(1/x)
ln(L) = lim (1/x) * ln(1 + x/2)
Theorems
L'Hopital's Rule
Logarithmic Approximation for Limits
Suitable Grade Level
Grades 11-12, Early College