Math 1304 Calculus I 2.4 – Definition for Limits

Recall Notation for Limits 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 a x f(x)L

How close? Measuring distance: argument and value. a x 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

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 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 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

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