Question

What are the asymptotes for `f(x) = 3 + log(x+2)`?

Answer 1

For one, the argument of a log-function must be positive.

Explanation:

This means `x> -2` and `x=2` is the corresponding (vertical) asymptote, because you can get as near to 2 as you want.
In "the language" we say:

`lim_(x->2+) f(x)= -oo`

There is no horizontal asymptote, as the log function will grow with `x` getting larger, even though the growth is slow, or:

`lim_(x-> oo) f(x) = + oo`
graph{3+log(x+2) [-5.67, 14.33, -3.32, 6.68]}