# Indeterminate Form

• Sep 28th 2009, 12:57 PM
soma
Indeterminate Form
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.
• Sep 28th 2009, 11:10 PM
mr fantastic
Quote:

Originally Posted by soma
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.

Actually the answer is $\displaystyle e^{7 + x}$.

You have $\displaystyle \lim_{n \rightarrow + \infty} \left( 1 + \frac{7 + x}{n}\right)^n = \lim_{n \rightarrow + \infty} \left( 1 + \frac{a}{n}\right)^n = e^a$.

(The last one is a standard limit and should be known).
• Sep 29th 2009, 04:59 AM
soma
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?
• Sep 29th 2009, 05:18 AM
Jester
Quote:

Originally Posted by soma
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)

$\displaystyle \lim_{n \to + \infty} \left( 1 + \frac{1}{n}\right)^n = e$
set $\displaystyle n = \frac{m}{a}$ noting that $\displaystyle n \to \infty$ gives $\displaystyle m \to \infty$ so
$\displaystyle \lim_{m \to + \infty} \left( 1 + \frac{a}{m}\right)^{m/a} = e$
$\displaystyle \left(\lim_{m \to + \infty} \left( 1 + \frac{a}{m}\right)^{m/a} \right)^a = e^a$.