Math 1304 Calculus I

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? (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

Rules for Distance • If a and b are real number |a-b| is the distance between them. • The distance between two different numbers is positive. • If the distance between two numbers is zero, then they are equal. • If a, b, c are real numbers, then |a - c| ≤ |a - b| + |b - c|

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 L+ f(x) L L- a- a x a+

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

Math 1304 Calculus I