Prove that $ \lim_{x \rightarrow 0}f(x)=L \leftrightarrow \lim_{x \rightarrow 0}[f(x)-L]=0 $

Question

Let $I \subset \mathbb{R}$ be an open interval, $c \in I$, and $f:I-\left \{ c \right \} \rightarrow \mathbb{R}$ be a function with $L \in \mathbb{R}$. Using only the definition of limits, prove that, $$ \lim_{x \rightarrow 0} f(x)=L \leftrightarrow \lim_{x \rightarrow 0}[f(x)-L]=0 $$

Answer 1

The crucial observation is that $|f(x)-L|<\epsilon$ is equivalent to $|(f(x)-L)-0|<\epsilon$.