Download One Limit Flowchart Establishing whether lim f(x) exists Two

One Limit Flowchart
Establishing whether x→a
lim f (x) exists
Start with a
function f (x)
and an x-value a
Does lim− f (x) exist?
x→a
yes
Does lim f (x) exist?
x→a+
no
yes
no
Is lim f (x) = lim f (x)?
x→a−
x→a+
yes
no
lim f (x) exists.
lim f (x) does not exist.
x→a
x→a
Two Continuity Flowcharts
Establishing that f (x) is continuous at x = a flowchart
Is f (a) defined? In
other words, is a in
the domain of f ?
yes
Start with a
function f (x)
and an x-value a
Does lim f (x) exist?
x→a
no
yes
no
Is lim f (x) = f (a)?
x→a
yes
f is continuous at a.
no
f is not continuous at a.
Using continuity to compute x→a
lim f (x) flowchart
Start with a
function f (x)
and an x-value a
Is f continuous at a?
no
huh?
yes, because someone trustworthy told me so
(or I used Theorem 7 on page 123)
If you are asked to evaluate lim f (x), you can
x→a
replace it with f (a).
Notation and terminology folow J. Stewart. Calculus: Early Transcendentals (7th ed.) as closely as possible.
E. Kim