Suppose I have some function $f(x)$ and some other function $g(x) = f(x-c)$.
Is is true that if $\lim_{x\to -c}f(x)$ exists, then $\lim_{x \to 0} g(x)$ exists and
$$\lim_{x\to -c}f(x) = \lim_{x \to 0} g(x)$$
?
It seems to make intuitive sense to me, as shifting a function doesn't seem like it should change the limit as long as we shift the value $x$ is approaching accordingly.