Understand the mathematics of continuous change.

Calculus is the mathematical study of things that change: cars accelerating, planets moving around the sun, economies fluctuating. To think about these evolving amounts, another arrangement of apparatuses – analytics – was created in the seventeenth century, always adjusting the course of math and science.

functional analytics experience that any yearning researcher, specialist, or mathematician needs.

Limits to Infinity

Sometimes we will not work something out directly … but we will see what it should be as we meet up with and closer!

Example:

(x2 − 1)(x − 1)

Let’s work it out for x=1:

Now 0/0 may be a difficulty! we do not really know the worth of 0/0 (it is “indeterminate”), so we’d like differently of answering this.

So rather than trying to figure it out for x=1 let’s try approaching it closer and closer:

Example Continued:

x — (x2 − 1)(x − 1)

0.5— 1.50000

0.9— 1.90000

0.99— 1.99000

0.999— 1.99900

0.9999— 1.99990

0.99999— 1.99999

… …

Now we see that as x gets on the brink of 1, then (x2−1)(x−1) gets on the brink of 2

We are now faced with a stimulating situation:

When x=1 we do not know the solution (it is indeterminate)

But we will see that it’s getting to be 2

We want to offer the solution “2” but can’t, so instead mathematicians say exactly what’s happening by using the special word “limit”

The limit of (x2−1)(x−1) as x approaches 1 is 2

And it’s written in symbols as:

limx→1 x2−1x−1 = 2

So it’s a special way of claiming , “ignoring what happens once we get there, but as we meet up with and closer the solution gets closer and closer to 2”

As a graph it’s like this:

So, in truth, we cannot say what the worth at x=1 is.

But we will say that as we approach 1, the limit is 2.

It is like running up a hill then finding the trail is magically “not there”…

… but if we only check one side, who knows what happens?

So we’d like to check it from both directions to make certain where it “should be”!

Example Continued

So, let’s try from the opposite side:

x (x2 − 1)(x − 1)

1.5 2.50000

1.1 2.10000

1.01 2.01000

1.001 2.00100

1.0001 2.00010

1.00001 2.00001

… …

Also heading for two , so that’s OK

Quick Summary of Limits

Sometimes we will not work something out directly … but we will see what it should be as we meet up with and closer!

Example:

(x2 − 1)(x − 1)

Let’s work it out for x=1:

Now 0/0 may be a difficulty! we do not really know the worth of 0/0 (it is “indeterminate”), so we’d like differently of answering this.

So rather than trying to figure it out for x=1 let’s try approaching it closer and closer:

Example Continued:

x (x2 − 1)(x − 1)

0.5 1.50000

0.9 1.90000

0.99 1.99000

0.999 1.99900

0.9999 1.99990

0.99999 1.99999

… …

Now we see that as x gets on the brink of 1, then (x2−1)(x−1) gets on the brink of 2

We are now faced with a stimulating situation:

When x=1 we do not know the solution (it is indeterminate)

But we will see that it’s getting to be 2

We want to offer the solution “2” but can’t, so instead mathematicians say exactly what’s happening by using the special word “limit”

The limit of (x2−1)(x−1) as x approaches 1 is 2

And it’s written in symbols as:

limx→1 x2−1x−1 = 2

So it’s a special way of claiming , “ignoring what happens once we get there, but as we meet up with and closer the solution gets closer and closer to 2”

As a graph it’s like this:

So, in truth, we cannot say what the worth at x=1 is.

But we will say that as we approach 1, the limit is 2.