My friend, who doesn't have access to the internet right now, needs help with this question...

Prove, using the precise definition, that $\displaystyle lim x->5 $ $\displaystyle \frac {1} {x - 5}$ does not exist

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- Nov 4th 2009, 11:25 AMdancecubedProof 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 $\displaystyle lim x->5 $ $\displaystyle \frac {1} {x - 5}$ does not exist** - Nov 4th 2009, 12:28 PMBruno J.
Well there are many precise definitions.

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