I am supposed to check the following function for continuity: $$h(x) = \frac {x^5}{x+5}$$
Knowing that the limits approaching from both sides have to be equal for the function to be continuous, I have no idea what to use for the left or the right side.
For example an exercise after this looks like this:
$$f(x)=\begin{cases}X/3, & x<=1\\ (x+1)^2 -11/3 & x>1\end{cases}$$
And I could solve it since I know X = 1 and had different functions for the limits on every side.
How should I approach this? I graphed it and it looks like continuous.
Definition : A function is continuous in its domain if it is continuous in every point of it.
The function $h(x) = \frac{x^5}{x+5}$ is defined on $D_h = \mathbb R \setminus\{-5\}$, since the denominator must be different than zero. It involves the quotient (or multiplication to be more rigorous) of polynomial terms, which are continuous and thus $h(x)$ is continuous in $D_h$.
The difference with the second one : The second one is indeed continuous in every point of its branches, but in order to be globally continuous you need to check continuity at the specific point which the branch changes. If and only if it is continuous there, it will be continuous globally as the branches are continuous in their respective domains. This is done, as you correctly said, by the limits approach.