Question

Analysis

"Let f :(a,b) -> R be a real-valued function on the open interval (a,b) and let \[c \in (a,b)\] .
(a) Write down the precise definitions in terms of \[\epsilon \] and \[\delta \] of the left and right limits
\[\lim_{x \rightarrow c+0} f(x)\]
\[\lim_{x \rightarrow c-0} f(x)\]
(b) Write down the precise definitions of the left and right derivatives of the function f at the
point 'c'.

Any help would be greatly appreciated, I am not familiar with left and right limits.

Answer 1

Drop the absolute value sign corresponding to the delta neighborhood in the typical epsilon delta definition and adjust the inequality accordingly, for example for the left handed limit, x is always going to be smaller then c, so we have if f approaches L then for any epsilon greater then zero there exists a delta so that for all
$$0<c−x<\delta\implies |f(x)−L|<ϵ$$

Answer 2

okay, I understand that, but what does it actually mean by 'left' and 'right' limits?

Answer 3

@sarahusher