Limits and Continuity
In calculus, we often ask what value a function is approaching for a given x value.
Thus, we want to know what the limit of the function is as we approach that x-value.
Formally, a limit is defined as follows:
Definition of a Limit:
Let f(x) be a function defined on an interval that contains xa
=
. Then,
lim ( )
xa
f
xL
→
=
if for every 0
ε
>, there exists a 0
δ
>such that
()fx L
ε
−<
whenever xa
δ
−
<
When I first saw the above definition, I didn’t know what it meant. It took me a while
to fully understand what it means. The definition says supposing the limit exists, then we
can set the distance between the function and the limit small (less than
ε
), by finding a
corresponding
δ
that will make it happen. A picture really helps to explain this.
Assuming the limit, L, exists, we can
draw two horizontal lines, one at L
ε
− and
the other at L
ε
+
. Now we can draw two
vertical lines, one at a
δ
−
and the other at
a
δ
+
.
a
L
a+δ
L+ε
a−δ
L−ε
Any x-value in the pink area will be
closer to a than either a
δ
−or a
δ
+. This
means that xa
δ
−
<.
For that x-value, the corresponding f
value will be in the orange region, whic
is the intersection of the pink and light
yellow areas. That implie
(x)
h
s()fx L
ε
−<
.
Now, keep repeating this process, continually choosing a smaller and smaller
ε
. Each
time, find a corresponding
δ
value. So long as the limit exists, we can continue
indefinitely, each time getting a bit closer to the actual value L as x gets closer to
a.
Notice that we never say tha t xa
=
. It is tempting to want to plug in a for x, but that
d be would not be the limit, that woul f(a). But wait, you might say, “I can look at the
picture and see that f(a) = L, so why can’t I just plug in the value and get my answer.
Why do I need to bother with
δ
and
ε
?”
