Calculus:Limits

$\lim _{x\to a}f(x)=L,$

When you see this equation it means you are trying to let x approaches a.

You may have a question why can't we just write it as

$f(a)=L,$

some times we can't tell what is F(a) equals.

example 1

$\lim _{x\to a}f(x)={x^{2}-1 \over x-1}$

But If a = 1 then you will get

$({1^{2}-1 \over 1-1})^{2}=({0 \over 0})^{2}$

Denominator can't be 0 so it is undefined at that point.

But in the graph if a = 1 it looks like it is than f(x) = 4

So the logic of the limit is approaching to not equal to. (what's the difference?)

it will be like this when we say x=1 then x is on.

But if we say $x\to 1$ then it could be 1.00000....0001 or 9.9999....999, x will not be 1 it will just be close to one.

So looking at the graph you may see h it is mean the rate of change of x most of the time we will like h or dx approaches to 0.