Solving the Limit Problem: (lim_{x to infty} frac{e^x}{x^{50}}) with LH?pitals Rule and Taylor Series Expansion

In this article, we explore a complex limit problem, (lim_{x to infty} frac{e^x}{x^{50}}). We discuss various mathematical methods including L'H?pital's Rule and Taylor Series Expansion to evaluate the limit and understand the underlying concepts.

Understanding the Limit

The limit we want to solve is given by:

[lim_{x to infty} frac{e^x}{x^{50}}]

It is known that as x approaches infinity, the exponential function e^x grows much faster compared to any polynomial function such as x^{50}. Therefore, the limit will diverge to positive infinity, as seen in the solution provided.

L'H?pital's Rule Approach

L'H?pital's Rule is a useful method for evaluating limits of the form (frac{infty}{infty}) or (frac{0}{0}). In this case, we apply L'H?pital's Rule 50 times to simplify the expression:

First, let's write the limit in an indeterminate form (frac{infty}{infty}). Apply L'H?pital's Rule 50 times by differentiating the numerator and the denominator 50 times. After 50 applications of L'H?pital's Rule, we get:

[lim_{x to infty} frac{e^x}{50!}]

Since the numerator remains (e^x) and does not change after differentiation, and the denominator now is a constant, the limit simplifies to (infty).

Taylor Series Expansion

Another approach to evaluate the limit is through Taylor Series Expansion. Recall that the Taylor series for (e^x) around 0 is:

[e^x sum_{i0}^{infty} frac{x^i}{i!}]

We can write the limit as follows:

[frac{e^x}{x^{50}} sum_{i0}^{infty} frac{x^{i-50}}{i!}]

Choose any (c > 0), then for all (b > c), we have:

[x 51!b Rightarrow frac{e^x}{x^{50}} frac{x}{51!} frac{51!b}{51!} b Rightarrow frac{e^x}{x^{50}} geq b forall x geq 51!c]

Therefore, by definition, the limit as (x to infty) is (infty).

Generalization

This approach can be generalized for a function (f(x) alpha frac{e^{lntx}}{x^n}) with (alpha > 0) and (t

[lim_{x to infty} f(x) alpha lnt^{infty} lim_{u to infty} frac{e^u}{u^n} infty]

Conclusion

In summary, we have explored the limit problem (lim_{x to infty} frac{e^x}{x^{50}}) using L'H?pital's Rule and Taylor Series Expansion. Both methods confirm that the limit is positive infinity. These techniques provide valuable tools for evaluating complex limits and understanding the behavior of functions as (x) approaches infinity.

Keywords:L'H?pital's Rule, Taylor Series, Exponential Growth