Limits

Can anyone help me. I suck at limits

here's the problem

Lim  (x-5)/(x^2-25)
x -> 5

>>9


I don't understand why that implies that limit must be the same on both sides.

Surely the function

f(x) = 1/((x+5) 0<x<5
f(x) = 1 + 1/(x+5) 5<=x<inf

this has the property that for a small neighbourhood around x=5 sgn(f(x))=1 but the limit as x->5 doesn't exist.


I don't get what point you're trying to make by bringing the sign function into this.