Question

If a function f(x) is continuous, then what can be said about the value of the following limit

lim-->2 [f(x) +4x]

Answer 1

I figured this out this way but am not sure it's right.

f(2) + 4(2) = f(2) + 8. From this we can tell that it exists, but I'm not sure why.

Answer 2

If f(x) is continuous (on ℝ), then f(2) must exist. Also, it is known that polynomials are continuous on ℝ.
Within these constraints, limits are additive, so
Lim x->2 f(x)+4x
=Lim x->2 f(x) + Lim x->2 4x
=f(2) + 4(2)
=f(2) + 8,
as you have found.