First we need to state this precise definition of a limit of a function f: X to RR:
lim_(x to x_0) f(x) = a in RR iff
iff forall epsilon > 0 exists delta > 0 forall x in X : |x-x_0| < delta => |f(x)-a| < epsilon
From a geometrical point of view, this means that saying this limit takes the value a is the same thing as saying that if x is near x_0 , then f(x) has to be near a, and this has to be valid no matter how small we consider the distance between f(x) and a.
For f(x) = 3x+5, the proof that lim_(x to 10) f(x) = 35 is very simple.
Given epsilon > 0, take delta = epsilon/3. Then, using the triangle inequality:
|x-10| < delta => |3x+5-35| = |3x -30| = 3 |x-10| < 3 delta = 3 epsilon/3 = epsilon iff |3x+5-35| < epsilon
This proof can be generalized to any function f that fulfils the condition:
exists k geq 0 forall x_1, x_2 in X : |f(x_2)-f(x_1)| leq k |x_2 - x_1|
Functions of this kind are caled Lipschitz continuous functions.
