# Math Help - Show that f(x) = 1/x is continuous at x=5.

1. ## Show that f(x) = 1/x is continuous at x=5.

Hi, I understand thus but I'm struggling with the proof.

I have so far:

f(5) = 1/5

so we need to show that for all positive epsilon(E) such that that there exists positive delta(d) s.t |(1/x) - 1/5)| < E for all x satisfying |x-5|<d

Then when I searched some threads on here I saw something that I think might help, but this is what it seemed to imply anyway :

Now |(1/x) - (1/5)| = |x-5| / |x||5|
However d>0 so when |x-5| = (1/2).|5| by the triangle law we would have
|x|=> 5-|x-5| >(1/2).|5|

therefore d <= (1/2) . |5|

I don't really see where the bit in red came from, which is why I'm not sure what I have found is correct, and I am also unsure how to finish off the proof. Any help would be great thank you

2. If $|x-5|<1$ then $\left| {\frac{1}{{5x}}} \right| < 1$.
So let $\delta = \min \left\{ {1,\varepsilon } \right\}$.