Let's learn about limit in A-Level Maths, which tell us how things turn out when a function reaches a certain value or towards infinity.

**Approaching...**

if x=1, it becomes 0 divided by 0.

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

Now we see that as x gets close to 1, then (x^{2}−1) /(x−1) gets close to 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 *(x*^{2}*−1)* /*(x−1)* as x approaches 1 is 2

And it is written in symbols as:

As a graph it looks lik the above:

So, in truth, we cannot say what the value at x=1 is. But we can say that as we approach 1, the limit is 2.

**Different from different sides**

How about a function f(x) with a "break" - 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"

**Approaching Infinity**

What is the value of **1/∞** ?

### We Don't Know!

Maybe we could say that *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 *1/***∞** is known to be undefined.

we can see that as x gets larger, *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

*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 *1/* **x** as x approaches Infinity is 0

Drafted by Eunice (Maths)

Reference

https://www.mathsisfun.com/calculus/limits.html