Introduction to Limit Calculations Involving Exponential Functions

Understanding the behavior of mathematical limits, especially as variables approach infinity, is a fundamental concept in calculus. This article will delve into the method of finding the limit of ( left(frac{x-2}{x^2}right)^{x^1} ) as ( x ) approaches infinity, focusing on a step-by-step approach that involves simplification and the use of exponential functions.

Step-by-Step Approach to Calculating the Limit

Let's start by simplifying the given expression ( left(frac{x-2}{x^2}right)^{x^1} ).

Initial Simplification

We first rewrite the fraction inside the limit:

$frac{x-2}{x^2} frac{1 - frac{2}{x}}{1 frac{2}{x}}$. As ( x ) approaches infinity, ( frac{2}{x} ) approaches 0, simplifying the fraction to 1. However, this simplification is not sufficient for further evaluation.

Revisiting the Expression

To proceed, we express the limit in terms of the natural exponential function ( e ). We rewrite the limit as:

(lim_{x to infty} left(frac{x-2}{x^2}right)^{x^1} lim_{x to infty} e^{x^1 ln left(frac{x-2}{x^2}right)})

Logarithmic Transformation

Next, we compute ( ln left(frac{x-2}{x^2}right) ):

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

We can further simplify this using the property of logarithms: ( ln(a) - ln(b) ln left(frac{a}{b}right) ) and the Taylor expansion for small ( u ), where ( ln(1 - u) approx -u ) as ( u ) approaches 0. Here, we can approximate:

(ln left(1 - frac{4}{x^2}right) approx -frac{4}{x^2})

Substituting and Simplifying

Substituting this back into our expression for the limit:

(x^1 ln left(frac{x-2}{x^2}right) approx x^1 left(-frac{4}{x^2}right) -frac{4}{x})

As ( x ) approaches infinity, ( -frac{4}{x} ) approaches 0, leading to:

(lim_{x to infty} e^{x^1 ln left(frac{x-2}{x^2}right)} e^{-4})

Thus, the final result is:

(lim_{x to infty} left(frac{x-2}{x^2}right)^{x^1} e^{-4})

Conclusion

The value of the limit as ( x ) approaches infinity is (e^{-4}). This problem demonstrates the importance of using exponential functions and properties of logarithms to simplify and solve complex limit problems.

Keywords: limit calculation, exponential function, Taylor expansion, infinity limit