Proving the limit of a function

Oct 2009
Aberystwyth, Wales
I can't get my head around these types of questions i know how to prove a limit you use \(\displaystyle |f(x) - L| < \epsilon\) but the question is use the (N, ε) method prove \(\displaystyle lim_{x \rightarrow \infty} \frac{7n + 3}{8n + 5} = \frac{7}{8}\)

I've read though the notes ive taken on this but still dont get it id be really grateful for some pointers :)
Last edited:
May 2010
Los Angeles, California
You have to show that for any \(\displaystyle \epsilon > 0\), there exists an \(\displaystyle N\) (depending on \(\displaystyle \epsilon\) ) such that whenever \(\displaystyle n > N\) we have that \(\displaystyle \Biggl| \frac{7n + 3}{8n + 5} - \frac{7}{8}\Biggr|<\epsilon\).

First put \(\displaystyle \frac{7n + 3}{8n + 5} - \frac{7}{8}\) over a common denominator and simplify. You should then be able to get \(\displaystyle N\) from the result.