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

Jun 13, 2017

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 $f \left(x\right) \le g \left(x\right) \le h \left(x\right)$ and ${\lim}_{x \to {x}_{0}} f \left(x\right) = {\lim}_{x \to {x}_{0}} h \left(x\right) = L$ then ${\lim}_{x \to {x}_{0}} g \left(x\right) = L$

3) If $f \left(x\right) \ge g \left(x\right)$ and ${\lim}_{x \to {x}_{0}} g \left(x\right) = \infty$ then ${\lim}_{x \to {x}_{0}} f \left(x\right) = \infty$

4) If ${\lim}_{x \to {x}_{0}} g \left(x\right) = 0$ and $f \left(x\right)$ is bounded, then ${\lim}_{x \to {x}_{0}} f \left(x\right) g \left(x\right) = 0$