1. Limits and Differentiation

by M. Bourne

To understand what is really going on in differential calculus, we need to have an understanding of limits.

Limits

Limits as x Approaches a Particular Number

Sometimes, finding the limiting value of an expression means simply substituting a number.

Example 1:

Find the limit as t approaches 10 of the expression P = 3t + 7.

Answer

We write this using limit notation as:

In this example there is no complication - we simply substitute and write

There is no complication because f(t) = 3t + 7 is a continuous function.

But there are cases where we cannot simply substitute like this.

Example 2:

We know that x cannot equal 3 in the following expression (because we cannot have a denominator equal to zero):

However, we can see that the function approaches a particular value as x approaches 3 from the left:

x	2.5	2.6	2.7	2.8	2.9
f(x)	3.5	3.6	3.7	3.8	3.9

Continuing, we get closer and closer to x = 3:

x	2.9	2.92	2.94	2.96	2.97	2.98	2.99
f(x)	3.9	3.92	3.94	3.96	3.97	3.98	3.99

Likewise, approaching x = 3 from the right gives the same limit value:

x	3.5	3.1	3.01	3.00001
f(x)	4.5	4.1	4.01	4.00001

We note that the function value is getting closer and closer to 4.

We write:

NOTE: We could have evaluated this limit by factorising first:

Answer

CAUTION: The factorising process is only possible in this example because we have: x ≠ 3.

This is a typical problem in the study of introductory limits. It appears to be a bit silly, in that we could have factored it, cancelled and substituted x = 3 like we just saw. But the example is important for the concept that there is no actual value of the function when x = 3, but if we get really, really close to 3, the function value is really close to some value (4, in this case).

Limits as x Approaches 0

We must remember that we cannot divide by zero - it is undefined.

But there are some interesting, and important, limits where there is a limiting value as x approaches 0 and where it would appear that we have a 0 denominator.

Example

Find the limit as x approaches 0 of

Answer

Notice we cannot just substitute 0 because is undefined.

There is no algebraic process to find this limit. We can substitute values of x which get closer and closer to 0 (from both the left side and right side) and conclude that

A way to check this is to graph it and see that indeed the limit as x gets closer to 0 is 1:

We cannot see it, but there is a "hole" at x = 0 in our graph.

Limits as x Approaches Infinity

Example 1:

Consider the fraction .

Answer

Clearly, if we take larger and larger values of x, the value of the fraction becomes smaller and smaller until it gets very close to 0. We say that "the limit of as x approaches infinity is 0".

We write this in mathematical notation as: .

In general:

And similarly,

We use these limits when evaluating limits of functions and it is especially useful in curve sketching.

Example 2:

Find the limit .

Answer

This time it is not so obvious what the limit value is. We could substitute larger and larger values of x until we could see what was happening (try 100, then 1000, then 1000000 and so on).

Or, we could rearrange the expression and use the fact that

to find the limiting value.

We divide throughout by x to get the expression in a form where we can evaluate it.

Notice that we cannot substitute ∞ into the fraction , because this does not make mathematical sense.

Please do not write math formula . It really upsets mathematicians.

Example 3:

Find

Let's see how this function looks in LiveMath.

LIVEMath

Here is the "normal" answer.

Answer

Joke

After explaining to a student about limits, I gave him the following example:

I tried to check whether he really understood that, so I gave him a different example.

The answer was:

Continuity and Differentiation

In this chapter we will be differentiating polynomials. But later we will come across more complicated functions and at times, we cannot differentiate them. We need to understand the conditions under which a function can be differentiated.

Earlier we learned about Continuous and Discontinuous Functions.

A function like f(x) = x³ − 6x² − x + 30 is continuous for all values of x, so it is differentiable for all values of x.

However, a function like math formula is not defined for x = 0 and x = 1.

It is discontinuous at those points. Hence, we cannot differentiate the function for those values.

math formula

Split Functions and Differentiation

We met Split Functions before in the Functions and Graphs chapter.

A split function is differentiable for all x if it is continuous for all x.

Example 1:

We met this example in the earlier chapter.

math formula

This function has a discontinuity at x = 1, but it is actually defined for x = 1 (and has value 1). It is differentiable for all values of x except x = 1, since it is not continuous at x = 1.

All of our functions in the earlier chapters on differentiation and integration will be continuous. In later chapters, we will see discontinuous functions, especially split functions. (see Fourier Series and Laplace Transforms)

We now move on to see how limits are applied to the problem of finding the rate of change of a function from first principles. This is the same as finding the slope of a tangent.