Solving the Limit: (lim_{x to infty} left(frac{x^4}{x-1}right)^x)

In this article, we explore the limit of the expression (lim_{x to infty} left(frac{x^4}{x-1}right)^x). This is a classic problem that involves an indeterminate form of 1^(infty). We will use both the Mean Value Theorem (MVT) and L'Hopital's rule to solve this problem and gain insight into the underlying mathematics.

Understanding the Problem

The expression (left(frac{x^4}{x-1}right)^x) can be seen as an indeterminate form of 1^(infty). Such forms often arise in limit problems where one part of the expression approaches 1 and the other grows without bound. To solve this problem, we can use a result that allows us to transform indeterminate forms into determinate forms, specifically using the exponential function.

Solving Using the Mean Value Theorem (MVT)

We can apply the Mean Value Theorem (MVT) to simplify the logarithm of the expression. The MVT states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists a point (c) in that interval such that:

(frac{f(x) - f(a)}{x - a} f'(c))

Here, we set (f(x) ln(x^4)) and we are interested in the interval between (x-1) and (x). Thus, the MVT implies the existence of a (c) between (x-1) and (x) such that:

[lnleft(frac{x^4}{x-1}right) frac{1}{c} left(x^4 - (x-1)^4right)]

As (x to infty), (frac{x^4 - (x-1)^4}{x}) approaches 5. This can be shown as:

[frac{x^4 - (x-1)^4}{x} frac{x^4 - (x^4 - 4x^3 6x^2 - 4x 1)}{x} frac{4x^3 - 6x^2 4x - 1}{x} approx 4x^2 - 6x 4]

For large (x), this simplifies to approximately 4x, and dividing by (x) gives 4.

Therefore, as (x to infty), (lnleft(frac{x^4}{x-1}right) approx 5). Exponentiating this, we get:

[left(frac{x^4}{x-1}right)^x e^{lnleft(frac{x^4}{x-1}right)^x} approx e^5]

Solving Using L'Hopital's Rule

Another approach is to use L'Hopital's rule, which is useful for resolving indeterminate forms like ( frac{0}{0} ) or ( frac{infty}{infty} ).

As we noted, the expression (left(frac{x^4}{x-1}right)^x) can be reinterpreted in a form where we can apply L'Hopital's rule. We start by rewriting the expression:

[ln y x lnleft(frac{x^4}{x-1}right) x lnleft(1 frac{5}{x-1}right)]

For large (x), (lnleft(1 frac{5}{x-1}right) approx frac{5}{x-1}). Thus:

[ln y approx x cdot frac{5}{x-1} 5 cdot frac{x}{x-1}]

As (x to infty), (frac{x}{x-1} to 1). Therefore:

[ln y to 5]

Exponentiating both sides gives:

[y to e^5]

Thus, the limit is:

[lim_{x to infty} left(frac{x^4}{x-1}right)^x e^5]

Significance and Generalizability

This proof is valuable for several reasons. First, it shows the origin of the number 5 within the expression. Second, it reduces the algebraic complexity compared to some other methods. Third, the method generalizes to other indeterminate forms of the 1^(infty) type. Additionally, this method is more intuitive as it leverages the power of the natural logarithm and exponential function.

The MVT and L'Hopital's rule are both powerful tools in calculus. Understanding how to apply these techniques can help solve a wide variety of limit problems. By studying such examples, one can enhance problem-solving skills and gain a deeper appreciation of the underlying mathematical principles.

Related Problems

Consider similar limit problems that use the same techniques:

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

These problems can be solved using similar approaches and techniques discussed in this article. Understanding the commonalities and variations in such problems can aid in solving them more efficiently.

Conclusion

The problem of finding (lim_{x to infty} left(frac{x^4}{x-1}right)^x e^5) demonstrates the power of both the Mean Value Theorem and L'Hopital's rule in resolving indeterminate forms. By using these techniques, we can understand the underlying structure of such limits and solve them with greater confidence. Whether you prefer the elegant simplicity of the MVT or the computational robustness of L'Hopital's rule, mastering these methods is crucial for any student of calculus.