Limits

The idea behind limits is to calculate what a problem will become as it approaches a an unanswerable point. For example an instance of this would answering something as it approaches infinity or an indeterminate answer. Sometimes we can't work something out directly, but we can see what it should be as we get closer to our answer.

Let's take a look at the following example:

\begin{equation} \frac{x^2 - 1}{x - 1} \end{equation}

For this example let's work it out for x=1:

\begin{equation} \frac{1^2 - 1}{1 - 1} = \frac{1 - 1}{1 - 1} = \frac{0}{0} \end{equation}

In this case 0/0 is indeterminate and can't be calculated so in a case like this we would use limits to determine what the answer should be. We do this by see what the x=1 is as it approaches our answer incrementally:

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

With this we can see that as x gets closer and closer to 1 our answer gets close to 2. Now this puts us in a predicament as we know that x=1 is unanswerable as it is indeterminate, but we can clearly see the answer is going to be 2. In this case mathematicians would say: "The limit of (x^2 - 1)/(x - 1) as x approaches 1 is 2."

This would be written as:

\begin{equation} \lim_{x\to1} \frac{x^2 - 1}{x - 1} = 2 \end{equation}

To definitively say that our answer is probably 2 we should also check the limit come from the other direction. In this example let's see what answers we get as we descend to x=1:

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

So in this case we can see that our answer is also heading to 2. This does bring up an interesting question though. What do we do when our answer is different coming from another direction?

To make things more simple let's just assume that we have a function: f(x) and we are trying to find the limit at point: a, but we can't answer this because we get a different answer depending on which direction we approach a. Let's assume for this example that we get the following:

• 3.8 from the left of a
• 1.3 from the right of a

In this case we would use the - or + signs to define each side of our limit:

\begin{equation} \lim_{x\to a^-} f(x) = 3.8 \hspace{1cm} \lim_{x\to a^+} f(x) = 1.3 \n \\\lim_{x\to a} f(x) = does \ not \ exist \end{equation}

Limits can be used even on non complex math problems, albeit a bit overkill:

\begin{equation} \lim_{x\to 10} \frac{x}{2} = 5 \end{equation}

A big place where limits come into play is when working with infinity, as infinity is more of a concept and than an actual number we can use in our day to day math. Take for instance:

\begin{equation} \frac{1}{\infty} \end{equation}

We have no way of actually getting an answer for this, but using limits we can see what the answer probably is:

x	1/x
1	1.00000
2	0.50000
4	0.25000
10	0.10000
100	0.01000
1000	0.00100
10000	0.00010
100000	0.00001

Now we can see that as x gets larger, our answer tends towards 0. We would write this limit as:

\begin{equation} \lim_{x\to\infty} \frac{1}{x} = 0 \end{equation}