Question

Prove that lim x--->c f(x)=L if and only if lim h-->0 f(h+c)=L

Answer 1

\[ \lim_{x\to c} f(x) = L \Leftrightarrow \forall \varepsilon > 0\ \exists \ \delta > 0 : \forall x\ (0 < |x - c | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon) \]
put \( x-c = h \) you get
\[ \lim_{x\to c} f(x) = L \Leftrightarrow \forall \varepsilon > 0\ \exists \ \delta > 0 : \forall x\ (0 < |h - 0 | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon) \]