Math 1304 Calculus I

2.4 – Definition for Limits

Recall Notation for Limits

Lxfax

)(lim

The reads: The limit of f(x), as x approaches a, is equal to L

Meaning: As x gets closer to a, f(x) gets closer to L.

Questions

• Can we make it more precise?

• Can we use this more precise definition to prove the rules?

Distance

• What do we mean by “As x gets closer to a, f(x) gets closer to L”.

• Can we measure how close? (We use distance.)

• What’s the distance?|x-a| is distance between x and a

|f(x)-L| is distance between f(x) and L

ax f(x) L

How close?Measuring distance: argument and value.

ax f(x) L

Distance of arguments =|x - a| Distance of values = |f(x)–L|

Can we make the distance between the values of f and L small by making the distance between x and a small?Turn this into a bargain: Given any >0, find >0 such that

|f(x)-L|< , whenever 0<|x-a|< .

Formal Definition of Limits

• Definition: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. We say that the limit of f(x) as x approaches a is L, if for each positive real number >0 there is a positive real >0 such that |f(x)-L|< , whenever 0<|x-a|< .

• When this happens we write

Lxfax

)(lim

Other ways to say it

• Given >0 there is >0 such that |f(x)-L|< , whenever 0<|x-a|< .

• Given >0 there is >0 such that 0<|x-a|< |f(x)-L|<

Definition in terms of intervals

• Given >0 there is >0 such that a- < x < a+ and x≠a L- < f(x) < L+

a a+a-

L L+L -

Picture

a a+a-

L

L+

L-

x

f(x)

Example

• Using the above definition, prove that f(x) = 2x+1 has limit 5 at x=2

• Method: guess and then verify. Work backwards: compute and estimate the

distance |f(x)-L| in terms of the distance |x-a|

Use the estimate to guess a delta, given the epsilon.

Nearby Behavior

• Note that if two functions agree except at a point a, they have the same limit at a, if it exists.

• Stronger result: If two functions agree on an open interval around a point a, but not necessarily at a, and one has a limit at a, then they have the same limit at a.

Proof of Rules

• Can prove the above rules from this definition.

• Example: the sum rule (in class)

• Note: we need the triangle inequality:|A+B| |A|+|B|

One-sided limits: Left

• Definition of left-hand limit

Lxfax

)(lim

if for each positive real number >0 there is a positive real >0 such that |f(x)-L| < , whenever a-<x<a.

One-sided limits: Right

• Definition of left-hand limit

Lxfax

)(lim

if for each positive real number >0 there is a positive real >0 such that |f(x)-L| < , whenever a<x<a+.

Limits of plus infinity

• Definition: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. We say that the limit of f(x) as x approaches a is , if for each real number M there is a positive real >0 such that f(x)>M, whenever 0<|x-a|< . When this happens we write

)(lim xfax

Limits of minus infinity

• Definition: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. We say that the limit of f(x) as x approaches a is -, if for each real number M there is a positive real >0 such that f(x)<M, whenever 0<|x-a|< . When this happens we write

)(lim xfax