A function f is defined by: $$f(x) = \begin{cases} x^4 + 1, \ \text{ if }x < 0,\\ \\ 0, \ \ \ \ \ \ \ \ \ \text{ if } x = 0,\\ \\ x^2 + 1, \ \text{ if } x > 0. \end{cases} $$
Find the limit as $x \to 0$
I know that the one-sided limits as $x\to 0^+$ and $x\to 0^-$ are both $1.$ However, the $f(x) = 0$ part confuses me a bit. My guess is that the limit does not exist, my reasoning being that because
$\lim_{x\to a} c = c$ , then
$\lim_{x \to 0} 0 = 0$
If this is not correct, then what bit of fundamental understanding am I missing?