limits and even-odd functions

Question

Is right to say the following?

• $$\lim_{x\to x_0}{f(x)}=k\overset{f: even}{\iff}\lim_{x\to x_0}{f(-x)}=k\overset{-x=t}{\iff}\lim_{t\to -x_0}{f(t)}=k\iff\lim_{x\to -x_0}{f(x)}=k$$

• $$\lim_{x\to x_0}{f(x)}=k\overset{f: odd}{\iff}-\lim_{x\to x_0}{f(-x)}=k\overset{-x=t}{\iff}\lim_{t\to -x_0}{f(t)}=-k\iff\lim_{x\to -x_0}{f(x)}=-k$$

Answer 1

Yes ! Your argumentation is fine !