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
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
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.