x -> 0
What is the indeterminate form of
lim n-> infinity (1 + 7/n + x/n)^n
If I let u = (7+x)/n, the problem becomes
lim n-> infinity (1+u)^((7+x)/u)
And the answer is e^7, but I don't understand the process of obtaining the answer.
for:
$\displaystyle \lim_{n \rightarrow + \infty} \left( 1 + \frac{a}{n}\right)^n = e^a$
How can I show the work on my homework assignment for that particular step? I know you say it's a standard limit, but I can't find any information about it online... Or is it just a matter of saying it's a standard limit?
Here's a site
http://en.wikipedia.org/wiki/E_(mathematical_constant)
As for your problem, from
$\displaystyle \lim_{n \to + \infty} \left( 1 + \frac{1}{n}\right)^n = e$
set $\displaystyle n = \frac{m}{a}$ noting that $\displaystyle n \to \infty $ gives $\displaystyle m \to \infty$ so
$\displaystyle \lim_{m \to + \infty} \left( 1 + \frac{a}{m}\right)^{m/a} = e$
so
$\displaystyle \left(\lim_{m \to + \infty} \left( 1 + \frac{a}{m}\right)^{m/a} \right)^a = e^a$.