Evaluating Complex Limits Involving Exponentials and Polynomials

In this article, we delve into the evaluation of complex limits, specifically involving exponentials and polynomials. We'll explore a detailed solution to the limit problem presented, which showcases the importance of logarithmic and exponential functions in advanced calculus. By understanding and applying these concepts, we can effectively handle similar problems.

Introduction to Limits and Polynomials

Limits are a fundamental concept in calculus, used to describe the behavior of functions as the input approaches a certain value. For limits involving variables raised to polynomial powers, understanding how to manipulate exponential and logarithmic expressions is crucial. The problem at hand, (lim_{{x to infty} left(frac{3x^2}{2x^2}right)^{2x^2 - 1}}), is a prime example of such a scenario.

Approaching the Problem

Let's break down the problem step by step:

Step 1: Simplifying the Base

The first step in solving the given limit is to simplify the base of the exponential expression. We start by noting that (frac{3x^2}{2x^2} frac{3}{2}) for all (x eq 0). However, for infinite limits, we need to be more precise:

((frac{3x^2}{2x^2})^{2x^2 - 1} (frac{3}{2})^{2x^2 - 1})

This simplifies our limit to (lim_{{x to infty}} (frac{3}{2})^{2x^2 - 1}).

Step 2: Logarithmic Transformation

Next, we apply the natural logarithm to both sides to make the problem more manageable:

Let (y (frac{3}{2})^{2x^2 - 1})

(ln y ln left((frac{3}{2})^{2x^2 - 1}right) (2x^2 - 1) ln left(frac{3}{2}right))

Now, we need to find (lim_{{x to infty}} (2x^2 - 1) ln left(frac{3}{2}right)):

(lim_{{x to infty}} (2x^2 - 1) ln left(frac{3}{2}right) infty times ln left(frac{3}{2}right) infty)

However, this approach isn't quite correct. Instead, we need to handle the expression ((1 frac{1}{2x^2})^{2x^2 - 1}).

Step 3: Substitution and Simplification

By letting (2x^2 1/t), we get:

(x^2 frac{1 - t}{2t})

(3x^2 frac{5t - 1}{2t})

(2x^2 frac{3t - 1}{2t})

The limit then becomes:

(lim_{{t to 0^ }} frac{ln left(5t - 1right) - ln left(3t - 1right)}{t})

Using the definition of the derivative, we recognize this as the derivative of (ln(5t - 1) - ln(3t - 1)) at (t 0):

Let (f(t) ln(5t - 1) - ln(3t - 1))

(f'(t) frac{5}{5t - 1} - frac{3}{3t - 1})

At (t 0), we have:

(f'(0) 5 - 3 2)

Therefore, the limit is (e^2).

Alternative Method: Using Substitution

Another approach involves using the substitution (2x^2 n), where (n to infty) as (x to infty). This transforms the original limit into:

(lim_{{n to infty}} left(1 frac{1}{n}right)^{2n - 3} lim_{{n to infty}} left(1 frac{1}{n}right)^{2n} left(1 frac{1}{n}right)^{-3})

Since (lim_{{n to infty}} left(1 frac{1}{n}right)^n e), we have:

(lim_{{n to infty}} left(1 frac{1}{n}right)^{2n} e^2) and (lim_{{n to infty}} left(1 frac{1}{n}right)^{-3} 1)

Hence, the limit is (e^2).

Conclusion

In conclusion, the limit (lim_{{x to infty} left(frac{3x^2}{2x^2}right)^{2x^2 - 1}}) evaluates to (e^2). This problem demonstrates the importance of logarithmic and exponential transformations in handling complex limits. Understanding these techniques is essential for solving similar problems in advanced calculus.