Calculating the Limit of ( lim_{x to infty} left( frac{x^3}{x-2} right)^{2x} ): Techniques and Insights

Introduction to Limits and Exponential Functions

When dealing with the limit of complex expressions, such as $lim_{x to infty} left( frac{x^3}{x-2} right)^{2x}$, understanding different techniques and properties of exponential functions becomes crucial. This article explores the evaluation of such a limit by breaking down the problem into simpler steps using algebraic manipulation and properties of logarithmic and exponential functions.

Simplifying the Fraction

The first step in evaluating the limit $lim_{x to infty} left( frac{x^3}{x-2} right)^{2x}$ is to simplify the fraction inside the limit. By dividing the numerator and the denominator by $x$, we get:

$frac{x^3}{x-2} frac{x^2 cdot x}{x left(1 - frac{2}{x}right)} frac{x^2}{1 - frac{2}{x}}$

For large values of $x$, the term $-frac{2}{x}$ approaches 0, simplifying the fraction to 1.

$lim_{x to infty} frac{x^3}{x-2} frac{x^2}{1 - 0} x^2$

However, this simplification does not directly help us with the limit as it is raised to the power of 2x. Therefore, we proceed by taking the natural logarithm of both sides to utilize the properties of logarithms and exponentials.

Applying the Natural Logarithm

Let $L lim_{x to infty} left( frac{x^3}{x-2} right)^{2x}$. Taking the natural logarithm of both sides, we get:

$ln L lim_{x to infty} 2x ln left( frac{x^3}{x-2} right)$

Using the property of logarithms, we can split the logarithm:

$ln left( frac{x^3}{x-2} right) ln(x^3) - ln(x-2)$

For large $x$, both logarithmic terms can be approximated as follows:

$ln(x^3) approx 3 ln x$ and $ln(x-2) approx ln x$

Therefore:

$ln left( frac{x^3}{x-2} right) approx 3 ln x - ln x 2 ln x$

Substituting this back into the limit, we get:

$ln L lim_{x to infty} 2x cdot 2 ln x lim_{x to infty} 4x ln x$

For large $x$, the term $4x ln x$ approaches infinity. Thus:

$ln L lim_{x to infty} 10 10$

Exponentiating both sides to find the original limit:

$L e^{10}$

Conclusion

The limit $lim_{x to infty} left( frac{x^3}{x-2} right)^{2x} e^{10}$. This demonstrates the power of using logarithmic and exponential functions to simplify and evaluate complex limits.

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