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AS/A-Level Mathematics - Intro to limits

Intro to limits

· A-Level Maths,limits,calculus,infinity,question analysis

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...

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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:

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Now we see that as x gets close to 1, then  (x2−1) /(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  (x2−1) /(x−1)  as x approaches 1 is 2 

And it is written in symbols as:

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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"

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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.

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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

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Drafted by Eunice (Maths)

Reference

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

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