Lecture 3 Intuitive Limits Now We Start Calculus

The Problem of Tangents

Tangents to a Circle Radius Tangent Line General line through P C = (h,k) Slope of radius =Slope of tangent =

Finding the Tangent Line at a Point to a circle of radius r, centered at C = (h,k) using algebra Point slope form: Suffices to find the slope Key – tangent meets the circle only in at the point P.

Find the points of Intersection Circle: Line Substitute for y

Illustrate what Happens for (h,k) = (0,0 ) LHS must equal Subtract and get So

Slope of Secant Connects (a, f(a)) with (a+h, f(a+h)) so slope is

Go through Similar Process With Some Functions Geometric Idea: Tangent at P meets the tangent line at only at P (near P)

Tangent to graph of at the point ( )

Problem: Such calculations are very difficult for more complicated functions – impossible for others. Need a new idea Don’t vary the slope of the general line – vary the other point The “secant line” should approximate the tangent line If one line “approximates” another then it’s slope should approximate that of the other line.

Tangent Line at (a,f(a)) Approximating Secant Line

Slope of the secant line is Idea is that as Q gets closer and closer to P (i.e. as h gets closer and closer to 0) the slopes of the secant lines get closer to that of the tangent line

We denote this Slope of the tangent line to graph of f at x = a is

Notation: If f is a function and a is in its domain then the slope of the tangent line to the graph of f(x) at the point (a,f(a)) is denoted f ‘ (a) so we write f ‘ (a) =

Calculate f ‘(4) if f ‘ (a) = Here a = 4 f ‘ (4) =

= 8

Calculate f ‘(a) for any a f ‘ (a) = = = ==

To do these must be able to calculate expressions of form Taken to (intuitively mean) : The value to which f(x) tends as x gets closer and closer to ( but never equals) a

Facts about Limits Limits may or may not exist – this limit does not exist There is an algebra of limits provided they exist

Some Basic Limits If c is a number and a is any number then If n is a number and a >0 then If f(x) and g(x) agree except at x = a then

Some Basic Non-Limits DOES NOT EXIST If c < 0 then DOES NOT EXIST If n > 0 then

If Here f(x) = x except at x = 0 so =

The basic limit theorems In general calculating a limit “from scratch” is difficult. The limit theorems allow us to calculate new limits from old without having to repeat what has already been done.

If exists and c is a number then Example:

If and both exist then Example: =

If and both exist then = = 12 example

If and both exist and is not zero then Example: = =

Easy Limits When f is defined by a single formula To calculate First see if you can calculate f(a). If this calculation produces a number then that is almost surely the correct answer. If f(a) has the form where c is not 0 then the limit does not exist If f(a) has the form then the limit may or may not exist. In this use algebra to find a function g(x) which agrees with f everywhere but at x = a. Then repeat the process with g(x).

Setting x = 2 gives Here = The last “ = “ holds except at x = 2 so g(x) = x+2 agrees with f(x) except at x = 2 therefore Substituting x = 2 now “makes sense” so the answer is 4 Example

Topic For Next Time: Continuity Read text book from Page 73 to 80 in Chapter 1 Assignment: Complete Homework 3 Start Homework 4