/sci/ - Science & Math

[Anonymous No.4070718]

Where would one find out more about converting a limit expression into e to a power of x.

An actual problem would be to convert lim [k->Infinity] of (1-2/k)^k to to e^x. Find out x. Solution to this problem is 1/e^2, but don't know why.

[Anonymous No.4070741]

whats the definition of e^x?
<div class="math"> e^x := \lim_{n \rightarrow \infty} \left(1 + \frac{x}{n}\right)^n. </div>
you can see here that x = -2 when you compare it with your thing.

[Anonymous No.4070815]

Okay, that part is now clear. One more thing, why is
<span class="math">
e^{\lim_{n \to \infty }{2m-1\over m+1}} = 2
[/spoiler]

[Anonymous No.4070841]

>>4070815
that's not right.
<div class="math"> \lim_{m \to \infty }{2m-1\over m+1}=2 </div>
though

[Anonymous No.4070845]

>>4070841
Yes, is this always the coefficient of the power if the power at numerator is the same as power in the denumerator?

[Anonymous No.4070875]

>>4070845
yes,for example
<div class="math"> \lim_{n \to \infty }{\frac{n+1}{n-1}}= \lim_{n \to \infty }{\frac{n+1}{n-1}\frac{\frac{1}{n}}{\frac{1}{n}}}= \lim_{n \to \infty }{\frac{1+\frac{1}{n}}{1-\frac{1}{n}}}= \lim_{n \to \infty }\frac{ \lim_{n \to \infty }(1+\frac{1}{n})}{ \lim_{n \to \infty }(1-\frac{1}{n})}=\frac{1+0}{1-0}=1</div>