Expometials Tend to Infinity Faster Than Any Power of x

  tends to infinity as  
  tends to infinity faster than any power of  
We can show this using l'Hospital's Rule repetedly, since  
\[lim_{x \rightarrow \infty} \frac{e^x}{x^n}\]
  is of indeterminate form.
\[\begin{equation} \begin{aligned} lim_{x \rightarrow \infty} \frac{e^x}{x^n} &= lim_{x \rightarrow \infty}\frac{e^x}{nx^{n-1}} \\ &= lim_{x \rightarrow \infty}\frac{e^x}{n(n-1)x^{n-2}} \\ &= \vdots \; \; \; \; \vdots \; \; \; \; \vdots \; \; \; \; \\ &= lim_{x \rightarrow \infty}\frac{e^x}{n!x^0} \\ &= \infty \end{aligned} \end{equation}\]
We can also show this with Mclaurin series.
\[\begin{equation} \begin{aligned} lim_{x \rightarrow \infty} \frac{e^x}{x^n} &= \frac{ \sum_{k=0}^{\infty} \frac{x^k}{k!}}{x^n} \\ &= \sum_{k=0}^{\infty} \frac{x^{k-n}}{k!} \end{aligned} \end{equation}\]

There are an infinite number of terms with positive power, each of which tends to infinity as  
  tends to infinity. Each term of the summation is positive for positive  
  so the sum and  
\[lim{x \rightarrow \infty} \frac{e^x}{x^n}\]
  tend to infity and the statement is true.

Add comment

Security code