Does the Statement $\lim_{f(x)\to a}k(x)$ Make Sense

Question

In a formal mathematics context does the statement $$\lim_{f(x)\to a}k(x)$$ where $f(x)\neq c$, where $c$ is a constant, make sense? For example does $$\lim_{x^2\to 0}x$$ make any sense in a formal and rigorous setting.

Answer 1

It's not quite standard notation, but the meaning is pretty clear: $$ \lim_{f(x) \to a} k(x) = B$$ would mean:

For every $\epsilon > 0$ there exists $\delta > 0$ such that whenever $0 < |f(x) - a| < \delta$, $|k(x) - B| < \epsilon$.

Answer 2

For every $c>0$ there exists $d>0$ such that $|f(x)-a|<d$ implies that $k(x)-l|<c$