# Limits *(An Introduction)*

## Approaching ...

Sometimes we can't work something out directly ... but we**can**see what it should be as we get closer and closer!

### Example:

\frac{(x^{2} − 1)}{(x − 1)}

Let's work it out for x=1:

\frac{(1^{2 }− 1)}{(1 − 1)} = \frac{(1 − 1)}{(1 − 1)} = \frac{0}{0}

Now 0/0 is a difficulty! We don't really know the value of 0/0 (it is "indeterminate"), so we need another way of answering this.

So instead of trying to work it out for x=1 let's try **approaching** it closer and closer:

### Example Continued:

x | \frac{(x^{2} − 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 close to 1, then \frac{(x^{2}−1)}{(x−1)} gets **close to 2**

We are now faced with an interesting situation:

- When x=1 we don't know the answer (it is
**indeterminate**) - But we can see that it is
**going to be 2**

We want to give the answer "2" but can't, so instead mathematicians say exactly what is going on by using the special word "limit".

The **limit** of \frac{(x^{2}−1)}{(x−1)} as x approaches 1 is** 2**

And it is written in symbols as:

*lim***x→1**\frac{x^{2}−1}{x−1} = 2

So it is a special way of saying,* "ignoring what happens when we get there, but as we get closer and closer the answer gets closer and closer to 2"*

As a graph it looks like this: So, in truth, we But we |

## Test Both Sides!

It is like running up a hill and then finding the path** is magically "not there"...**

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

So we need to test it **from both directions** to be sure where it "should be"!

### Example Continued

So, let's try from the other side:

x | \frac{(x^{2} − 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 2, so that's OK

## When it is different from different sides

How about a function **f(x)** with a "break" in it like this:

The limit does not exist at "a"

**We can't say what the value at "a" is**, because there are two competing answers:

- 3.8 from the left, and
- 1.3 from the right

But we **can** use the special "−" or "+" signs (as shown) to define one sided limits:

- the
**left-hand**limit (−) is 3.8 - the
**right-hand**limit (+) is 1.3

And the ordinary limit **"does not exist"**

## Are limits only for difficult functions?

Limits can be used even when we **know the value when we get there**! Nobody said they are only for difficult functions.

### Example:

*lim***x→10**\frac{x}{2} = 5

We know perfectly well that 10/2 = 5, but limits can still be used (if we want!)

## Approaching Infinity

Infinity is a very special idea. We know we can't reach it, but we can still try to work out the value of functions that have infinity in them.

### Let's start with an interesting example.

Question: What is the value of \frac{1}{∞} ? |

Answer: We don't know! |

### Why Don't We Know?

The simplest reason is that Infinity is not a number, it is an idea.

So \frac{1}{∞} is a bit like saying \frac{1}{beauty} or \frac{1}{tall}.

Maybe we could say that \frac{1}{∞}= 0, ... but that is a problem too, because if we divide 1 into infinite pieces and they end up 0 each, what happened to the 1?

In fact \frac{1}{∞} is known to be **undefined**.

### But We Can Approach It!

So instead of trying to work it out for infinity (because we can't get a sensible answer), let's try larger and larger values of x:

x |
\frac{1}{x} |

1 | 1.00000 |

2 | 0.50000 |

4 | 0.25000 |

10 | 0.10000 |

100 | 0.01000 |

1,000 | 0.00100 |

10,000 | 0.00010 |

Now we can see that as x gets larger, **\frac{1}{x}** tends towards 0

We are now faced with an interesting situation:

- We can't say what happens when x gets to infinity
- But we can see that
**\frac{1}{x}**is**going towards 0**

We want to give the answer "0" but can't, so instead mathematicians say exactly what is going on by using the special word "limit".

The **limit** of **\frac{1}{x}** as x approaches Infinity is** 0**

And write it like this:

*lim***x→∞**\frac{1}{x} = 0

In other words:

As x approaches infinity, then **\frac{1}{x}** approaches 0

*When you see "limit", think "approaching"*

It is a mathematical way of saying *"we are not talking about when x=∞, but we know as x gets bigger, the answer gets closer and closer to 0"*.

Read more at Limits to Infinity.

## Solving!

We have been a little lazy so far, and just said that a limit equals some value because it **looked like it was going to**.

That is not really good enough! Read more at Evaluating Limits.