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 f(x)=2x3x1f(x) = \frac{2^x}{3^x - 1} as xx \to \infty.

Step-by-Step Solution:

  1. Rewrite the function: f(x)=2x3x1f(x) = \frac{2^x}{3^x - 1} As xx \to \infty, both 2x2^x and 3x3^x grow exponentially, but 3x3^x grows faster because the base (3) is larger than 2.

  2. Factor the denominator to compare terms: Factor out 3x3^x from the denominator: f(x)=2x3x(113x)f(x) = \frac{2^x}{3^x(1 - \frac{1}{3^x})}

  3. Simplify the expression: f(x)=(2x3x)(113x)=((23)x)(113x)f(x) = \frac{\left( \frac{2^x}{3^x} \right)}{\left( 1 - \frac{1}{3^x} \right)} = \frac{\left( \left( \frac{2}{3} \right)^x \right)}{\left( 1 - \frac{1}{3^x} \right)}

  4. Analyze the behavior as xx \to \infty:

    • As xx \to \infty, (23)x0\left( \frac{2}{3} \right)^x \to 0 because 23<1\frac{2}{3} < 1.
    • Also, 13x0\frac{1}{3^x} \to 0, so the denominator approaches 1.
  5. Take the limit: limxf(x)=limx(23)x10=0\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{\left( \frac{2}{3} \right)^x}{1 - 0} = 0

Conclusion:

limx2x3x1=0\lim_{x \to \infty} \frac{2^x}{3^x - 1} = 0

Would you like further details or have any questions? Here are 5 related problems you might find interesting:

  1. Find limx5x6x+2\lim_{x \to \infty} \frac{5^x}{6^x + 2}.
  2. What is limx2x3x1\lim_{x \to \infty} \frac{2x}{3x - 1}?
  3. Compute limxexe2x1\lim_{x \to \infty} \frac{e^x}{e^{2x} - 1}.
  4. Evaluate limx3x+15x\lim_{x \to \infty} \frac{3^x + 1}{5^x}.
  5. Determine limxln(x)x\lim_{x \to \infty} \frac{\ln(x)}{x}.

Tip: When comparing exponential functions, the one with the larger base dominates as xx \to \infty.

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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