What does it mean graphically if a function has the following limit: lim→0ℎ(ℎ())=3?

Question

What would it mean if the following property was part of a graph?

$$\lim_{x\to 0} h(h(x))=3$$

My current understand is that it would be a horizontal asymptote at $y = 3$?

See graph

Answer 1

It says that when $x$ is very close to $0$, $h(h(x))$ is very close to $3$. One way this can happen (but not the only way) is that $\lim_{x \to 0} h(x)$ is some number $L$, and $\lim_{x \to L} h(x) = 0$. Thus when $x$ is very close to $0$, $h(x)$ would be very close to $L$, and when $x$ is very close to $L$, $h(x)$ would be very close to $0$. Thus in the case $L=2$, two pieces of the graph of the function might look something like this:

Answer 2

Not necessarily. Infinitely many functions of all shapes satisfy this.

For example, $y=2x+1$ satisfies this without any horizontal asymptotes. But $y=3+\frac{1}{x}$ satisfies this with asymptotes.