Question

When do we know that a limit of a given function exists?

Answer 1

In general you know that a given function has a limit only by determining its limit.

There are however some theorems ensuring a function has a limit under certain hypotheses, for example:

1) A monotonous function in an interval always has a limit

2) if `f(x) <= g(x) <= h(x)` and `lim_(x->x_0) f(x) = lim_(x->x_0) h(x) = L` then `lim_(x->x_0) g(x) = L`

3) If `f(x) >= g(x)` and `lim_(x->x_0) g(x) = oo` then `lim_(x->x_0) f(x) = oo`

4) If `lim_(x->x_0) g(x) = 0` and `f(x)` is bounded, then `lim_(x->x_0) f(x)g(x) = 0`