As we saw the derivative is defined in terms of limits. But what do we mean by limit exactly.

We'll  start by considering a graphical approach to limits.

Then we'll consider a rigorous definition of limit.

Finally we'll consider the computational aspects of limits.

We start with a discussion and animations defining one-sided limits.

The next 2 animations deal with the function f(x)  = | x² - 4 | if x ≠ 2 and f(2) =3.

The first animation is the graph being generated and the second shows the one- sided limits .

If lim _{x ->a} f(x) = f(a) we say a function is continuous at x = a .

This implies 3 things:

1. lim _{x ->a} f(x) exists

2. f(a) is defined

3. 1 = 2

This example shows a limit can exist at a point even if in fact  f(x)  is not continuous.

In fact if f(x)  = | x² -4 | if x ≠ 2 and f(2) is not even defined the limit  still exists.

The reason why this is so important is that the derivative is defined as a limit which invariably is 0/0 . The term (f(x+h) - f(x))/h  is not defined  when h = 0 however the fact the limit exists gives us the derivative and indeed Calculus itself.

Limits at  ±∞ 

The following 2 animations deal with lim_x->0 sin(1/x)  and lim_x->0 xsin(1/x).

In the first case as we zoom near 0 sin(1/x) oscillates at an increasingly rapid rate between -1 and 1 and hence approaches no limit. The reason is  as x → 0 1/x→∞ and sin(x) does not approach a limit at ∞.

However  xsin(1/x)  is squeezed between y = x and y = -x both of which approach 0 as x approaches 0 and therefore xsin(1/x) is squeezed to 0 as x goes to 0.

This is an example of a very useful tool - the squeezing theorem:

Thm:

If f(x) < h(x) < g(x) for all x in an open interval containing x =a and

 lim _x->a f(x) =  lim _x->a g(x) = L Then lim _x->a h(x) = L

So how do we actually Calculate Limits?

We have seen how to evaluate limits grahically. To use our rigorous definitions you have to already know what the limit is to proceed. This is important in making exact what we mean by terms such as "approaches"  or " nears" but doesn't really tell us how to calculate limits.

The download below are my Lecture notes on the computational aspects of limits.

The Derivative and one-dimensional motion

Two further examples.
In the first animation we see the freely falling object--an object is thrown upward under the fluence of gravity only. Note we also consider the difference between velocity and speed i.e. speed = |v(t)| . In the second animation we  see the same ideas for a particle moving horizontally

Mean Value Theorem

One  of the most important theorems in all of Calculus is

the mean value theorem. The mean value theorem simply stated

says that if we measure the average rate of change over an interval (a,b)

then at some point c in (a,b) the instantaneous rate of change, f ' (c),

is equal to the average rate of change over the interval a to b.

Geometrically we are saying at some point c the slope of the tangent

line is the same as the slope of the secant line through the points

(a,f(a)) and (b,f(b))

Formally If f(x) is continuous on [a,b] and differentiable on (a,b) then

there is a number c in (a,b) such that f ' (c) = [f(b) -f(a)] / [b-a)

The following animation demonstrates this for f(x) = sin(x)

Local Linearity and the Differential

Another way of characterizing differentiability is local linearity. As the next animation shows if f(x) is differentiable at a point we see that if you zoom in on a pt the function and its tangent line converge

The second animation shows at a point of non-differentiability there is no local linearity. The second animation is f(x) = |x| at x = 0.

For an explanation of a point of non-differentiability see the download A point of non-differentiability-lecture notes. It explains from the definition of derivative why there is no derivative at x = 0 for f(x) = |x|.


One application of this the differential - the complete lecture notes being provided below