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Numerade Educator

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Problem 73 Hard Difficulty

Prove that
$$ \displaystyle \lim_{x\to \infty} \frac{e^x}{x^n} = \infty $$
for any positive integer $ n $. This shows that the exponential function approaches infinity faster than any power of $ x $.

Answer

$$\lim _{x \rightarrow \infty} \frac{e^{x}}{x^{n}} \stackrel{\Perp}{=} \lim _{x \rightarrow \infty} \frac{e^{x}}{n x^{n-1}} \stackrel{\Perp}{=} \lim _{x \rightarrow \infty} \frac{e^{x}}{n(n-1) x^{n-2}} \stackrel{\Perp}{=} \cdots \stackrel{n}{=} \lim _{x \rightarrow \infty} \frac{e^{x}}{n !}=\infty$$

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Video Transcript

All right, we've got a question here that gives us the limit as X approaches infinity or e of X over X to the end. And we want to prove that this is true for all positive integers of end. First of all, if we know that we have all positive integers and then this will be infinity at the bottom on then e to the infinity would also go to infinities Say that we would get an intermediate type form of the type of infinity of infinity so we can apply it. Law hop hospitals rule, and that would be the limit. As X approaches infinity, eat the x x t n is equal to same thing as the limit. That box approaches affinity with the D Do you x x over the d deluxe next n on that comes out to be the moment as X approaches infinity for FX now n xcn okay. And if we do that, we would get in infinity over 30. Alright, once again, that is Theo Intermediate Form s. So what we're gonna do is we're gonna apply the law hospital who of another time, and we will keep And if we keep doing it. We get to the point where we realize that our our denominator is reaching infinity faster. Andi, excuse me. No, you would realize that our denominator, our limit will stay and enter in determinant form until the degree of excellent denominated becomes zero. Right? So it'll go on forever. So for example, if you take the lobby to all of this, you'll get when Mitch as X approaches infinity with a d d x e two x over DT x and next to the and Children end there. And that comes out to be when it x infinity to the X and multiplied by and minus one need to the n minus two, and we're still gonna be left with infinity over infinity. All right, now we're gonna use what's called applying the law. Hospitals rule now end times. And if we do that, then we would get limit as X approaches infinity each of the X and factorial. Okay, if we do that, then we have one factorial one over n factorial need to the infinity and that will be equal to infinity. Okay. And then here we can see that the exponential function approaches infinity faster than any power of X. All right, And we can prove that this function here is true. Um, for all fact, all positive values of it. All right, well, I hope that clarifies the question. Thank you so much for watching.