Developing the formal definition of a limit
Given a function f (x) that is defined around a point a (but not necessarily
defined at a) we want to develop a formal definition
lim f (x) = L
x→a
Key Concept
Intuitively, a limit exists if:
You can force the values of f (x) close to L by restricting your
function to smaller and smaller intervals around the point a.
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Developing the formal definition of a limit
We think that limx→1 f (x) = 2.
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Developing the formal definition of a limit
Force the values of f (x) to within
1
2
of L = 2
Red lines indicate this range.
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Developing the formal definition of a limit
Force the values of f (x) to within
1
2
of L = 2
We can achieve this by restricting our function to within the purple lines
around a = 1
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Epsilon
On the last slide we forced our function to within
the point a.
More generally think about replacing
denote > 0
1
2
1
2
of L = 2 around
with any value, which we
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Developing the formal definition of a limit
Suppose we can do this for any value > 0. Then as the values for grow
smaller (i.e 14 , 18 ...), the function is forced to converge on L
Figure: =
1
2
Figure: =
1
4
Figure: =
1
8
Red lines represent the values of January 9, 2017
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Formal Definition of a limit
We say that the limit of f (x) as x approaches a is L and write
lim f (x) = L
x→a
if for every > 0, there is a δ > 0 such that
if
0 < |x − a| < δ
then
|f (x) − L| < January 9, 2017
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Interpreting the formal definition of a limit
|f (x) − L| < :
tells us how close we forcing our function to be to L.
|f (x) − L| < ⇒
0 < |x − a| < δ:
⇒
− ≤ f (x) − L ≤ L − ≤ f (x) ≤ L + tells us the interval in which f (x) is within of L.
|x − a| < δ
⇒
⇒
−δ ≤ x − a ≤ δ
a−δ ≤x ≤a+δ
Note: 0 < |x − a| simply implies that x 6= a
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