Math Problem Statement
What is the limit of x as x approaches infinity of f(x)=(2^x)/(3^x-1)
Solution
We are tasked with finding the limit of as .
Step-by-Step Solution:
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Rewrite the function: As , both and grow exponentially, but grows faster because the base (3) is larger than 2.
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Factor the denominator to compare terms: Factor out from the denominator:
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Simplify the expression:
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Analyze the behavior as :
- As , because .
- Also, , so the denominator approaches 1.
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Take the limit:
Conclusion:
Would you like further details or have any questions? Here are 5 related problems you might find interesting:
- Find .
- What is ?
- Compute .
- Evaluate .
- Determine .
Tip: When comparing exponential functions, the one with the larger base dominates as .
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Math Problem Analysis
Mathematical Concepts
Limits
Exponential Functions
Asymptotic Behavior
Formulas
lim (x -> ∞) f(x) = lim (x -> ∞) (2^x) / (3^x - 1)
Theorems
Limit Comparison of Exponential Functions
Suitable Grade Level
Grades 10-12
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