Math Help

Limit of a Function (Continued)

Limits can be “one-sided” depending if x approaches a from a larger or from a

smaller value of a. For example:

If x approaches a from smaller values: lim f(x) = L₁

x →a^˗

If x approaches a from larger values: lim f(x) = L₂

x → a⁺

Note: A limit exists if and only if both corresponding one-sided limits exist and are

equal. That is:

lim f(x) = L, for some number L , if and only if, lim f(x) = lim f(x) = L

x → a x → a^˗ x → a⁺

Formal Definition of a Limit of a function, f(x), of one independent variable, x.

Let f be a function, f(x), defined in an open interval containing a (but not necessarily

at a itself), then

lim f(x) = L

x → a

if given an arbitrary number ε > 0, there is another number δ > 0, such that

0 < | x – a | < δ guarantees that | f(x) ˗ L | < ε .