MHB Understanding Limit Notation & Symbols: L & <=>

[tmt1]

Hello,

My professor wrote something on the board the other day and I forgot to ask after class.

$$\lim_{{x}\to{c}} f(x)$$ = L <=>$$\lim_{{x}\to{c^+}} f(x)$$ = L && $$\lim_{{x}\to{c^-}} f(x)$$ = L

What does mean by L and what does <=> mean?

 

[mathbalarka]

The $\LaTeX$ notation for <=> is $\iff$, the iff operation. $A \iff B$ means "if A then B and if B then A".

$L$ is simply the limit, the value of $\lim_{x \to c} f(x)$.

 

[Evgeny.Makarov]

$L$ is a variable that ranges over real numbers. The statement
\[
\lim_{{x}\to{c}} f(x)=L\iff \left(\lim_{{x}\to{c}^+} f(x)=L\text{ and }\lim_{{x}\to{c}^-} f(x)=L\right)
\]
whatever it means, has three variables: $f$, $c$ and $L$. Here $f$ ranges over functions from $\Bbb R$ to $\Bbb R$, while $c$ and $L$ range over $R$. If a statement has variables like this, it usually means that they are universally quantified, i.e., the claim is that the statement holds for all $f$, $c$ and $L$.

This particular statement says that the limit of $f$ is $L$ iff both the right-sided and the left-sided limits of $f$ are $L$.