Calculus:Limits

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

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

You may have a question, "why can't we just write it as $f(a)=L$?"

Sometimes we can't tell what 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 one.

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.

When looking at the graph, you will see h equals Δx. Δx means the rate of change of x. Most of the time we will like h or Δx approaches to 0.

Properties of Limits

1. Sum Rule: The limit of the sum of two functions is the sum of their limits : $\lim _{x\to c}(f(x)+g(x))=\lim _{x\to c}(f(x))+\lim _{x\to c}(g(x))$
2. Difference Rule: The limit of the difference of two functions is the difference of their limits : $\lim _{x\to c}(f(x)-g(x))=\lim _{x\to c}f(x)-\lim _{x\to c}g(x)$
3. Product Rule: The limit of a product of two functions is the product of their limits : $\lim _{x\to c}(f(x)*g(x))=\lim _{x\to c}f(x)*\lim _{x\to c}g(x)$
4. Constant Multiple Rule: The limit of a constant times a function is the constant times the limit of the function :

$\lim _{x\to c}(k*g(x))=k*\lim _{x\to c}g(x)$

1. Quotient Rule: The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero. $\lim _{x\to c}{(f(x) \over g(x))}={\lim _{x\to c}f(x) \over \lim _{x\to c}g(x)}$