# Proof that Limits doesn't exist - Help...

• Nov 4th 2009, 12:25 PM
dancecubed
Proof that Limits doesn't exist - Help...
My friend, who doesn't have access to the internet right now, needs help with this question...

Prove, using the precise definition, that $lim x->5$ $\frac {1} {x - 5}$ does not exist
• Nov 4th 2009, 01:28 PM
Bruno J.
Well there are many precise definitions.

Let $f(x)=\frac{5}{x-5}$. Suppose $\{x_n\}$ is a sequence of positive real numbers tending to 0. Then $\{5-x_n\}$ and $\{5+x_n\}$ both tend to 5 but $\{f(5-x_n)\}$ tends to $-\infty$ while $\{f(5+x_n)\}$ tends to $+\infty$. Therefore the limit does not exist.