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Problem 56 Hard Difficulty

Show that $ \displaystyle \lim_{n \to x} (1 + \frac {x}{n})^n = e^x $ for any $ x > 0. $

Answer

Therefore $\lim _{\lim _{\infty}}\left(1+\frac{x}{n}\right)^{n}=\lim _{m \rightarrow \infty}\left(1+\frac{1}{m}\right)^{m x}=\left[\lim _{m \rightarrow \infty}\left(1+\frac{1}{m}\right)^{m}\right]^{x}=e^{x}$
by Equation $6 .$

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Video Transcript

in this problem with us to show that limit hasn't purchased in 30 or plus exhort and power and is equal to B to the X 14 years. Zero Let's say that this function is function. Why? Which is a function of X Unless you like. This one has limit is engrossed in purity. Why physical X now the beauty of nature water, boat science we can rip this oneness is engrossing unity actual level Why physical to mention love eat x and that is evil ex. So if we could prove that limit off natural log off dysfunction as Anna purchasing through these X, then it means that we also pro this one all right? No, we have one plus so wrenched and powers while less tech natural Log off that financial level. Why would be Mitchell or 01 bus, excellent and and power. And that will be times natural law off horn plus X over. Now let's say that you're interested in him that as an effort, your student beauty that is in terms nature law off plus excellent. Been. And as an airport, you see on your TV with get infinity terms of zero and that doesn't really mean it makes sense. But if we write this limit as limit is that purchased in energy mention law off on, plus Excellent. You got it. If I went over and you would get 010 that is Indeterminant, then we could apply up, tells rule or are important thing here is that n is very evil. That is aversion to infinity. So what a final cultural meaning one. Taking the inevitable porter waiter and, you know, had nerve enough to take the oath and less to that limit. As Anna purchased immunity, we have limited d the original function numerator that is one over one plus excellent end times negative eggs or inscribe divided by the limit off function denominator. That is a negative or an expert. No, these will be positive. These will go away. And now we have limits. As I have purchased in 20 thanks. Derided by one plus X, we can order right this one US limit as in a purpose, security in terms of X divided by and X Now, as an approach Houston community, this is intent, your impunity. We can apply also one more time. So let's take give or numerator and denominator, It should give us limit as any purchased in unity X, divided by one. And that is it ex we just saw. That limit is an approaches infinity actual love of dysfunctional, purchased ex pain. Remember, that is what we were Well.