Question

How can I conceptualize limits?

Answer 1

I mean like on some problems there is \[\lim_{n \rightarrow \infty} \]
And I find this kind of hard to imagine, a number going from 1 and going on to infinity. I understand that it is getting infinitely close to it, but I was just wondering if anyone had some tips on how to think of it.

Answer 2

Let say you have that expression:
\[\lim_{n \rightarrow \infty} \frac{ 2x^2+x+5 }{ 3x^2-6 }\]

I think it like this: x gets larger and larger and terms with x^2 becomes much larger than others. For example if x is 1 million than x^2 would be 1 million times of terms with x. You can ignore them. And infinity is much more bigger than just a million. If I ignore the other terms I have just 2x^2/3x^2 which is equal to 2/3. Its about ignoring small things (or big things sometimes) when limit goes to infinity.

Answer 3

So if we had that problem, we would divide out the largest power in it, right?
\[\frac{ \frac{ 2x^2 }{ x^2 } + \frac{ x }{ x^2 } + \frac{ 5 }{ x^2 }}{ \frac{ 3x^2 }{ x^2 } - \frac{ 6 }{ x^2 } }\]
Which the terms that do not divide out evenly would be zeros, and those that do would be the number that y would be getting infinitely close to?

Answer 4

Yep. But first simplify it. Then you will find that:
\[\frac{ 2+1/x+5/x^2 }{ 3-6/x^2 }\]
Terms with 1/x and 1/x^2 get close to 0 as x goes to infiniity. However 2 and 3 do not change. Ignore other terms. That expression is equal to 2/3.

Answer 5

Thank you, sir.