1. Notion of limit

Hello,

for many years i have been doing derivatives, and its definition: lim f(x+h)-f(x) / h always seemed quite obvious to me, but then i started to think about it...
h->0

when we do derivatives through its definition, we usually arrive to something like (h(x+h))/h and we remove the h on the numerator and denominator, leaving x+h, when we replace h by 0 and get to x....but the question is, wont we be doing a violation when we place h by 0? If we condsidered the previous formula, replacing 0 by 0 would cause a 0/0 indetermination, which arises from the fact that the removal rule only applices if h!=0, so....what is that last result we get? The result of ignoring the fact that h cannot be 0 but at the same time assume it is 0?

2. Re: Notion of limit

Since derivatives are continuous already, we can put h = 0.

3. Re: Notion of limit

Then why did we use a rule that considered that h wont be zero?

4. Re: Notion of limit

Originally Posted by Kanwar245
Since derivatives are continuous already, we can put h = 0.
What make you say that. That is false.
If a function has a derivative at $\displaystyle x_0$ then the function is continuous at $\displaystyle x_0.$
We don't about the derivative at $\displaystyle x_0$.

Originally Posted by DarkFalz
Then why did we use a rule that considered that h wont be zero?
When evaluating $\displaystyle {\displaystyle\lim _{x \to {x_0}}}f(x)$ we never just let $\displaystyle x=x_0$.
That is a fundamental principle of limits. We must look for continuity.
When considering the difference quotient $\displaystyle \dfrac{f(x+h)-f(x)}{h}$ we hope that the $\displaystyle h$ divides out.
That can only be if $\displaystyle h\ne 0.$

5. Re: Notion of limit

But after that we place h as 0, so is it valid to place h as 0 or not?

6. Re: Notion of limit

Originally Posted by DarkFalz
But after that we place h as 0, so is it valid to place h as 0 or not?
If the that means you are not dividing by h.
What we are really saying, h is approximately zero, but not zero.
That is a very technical distinction but an important one.

7. Re: Notion of limit

If it is not zero, why do we replace h by 0 in the end?

8. Re: Notion of limit

Originally Posted by DarkFalz
If it is not zero, why do we replace h by 0 in the end?
WE DON'T.
We simply see what happens for values of h nearly zero.
Lazy teachers allow replacement. But it really is not correct.
If the difference quotient reduces to a continuous function of h, then it works.
It works, but technically not correct.

Because a derivative is a limit, we must follow the formal definition of limit.
In the definition of $\displaystyle {\displaystyle\lim _{x \to {x_0}}}f(x)$ it is required that $\displaystyle x\ne x_0.$

9. Re: Notion of limit

If reducing to a continuous function and replacing works, which is the other way to find the limit?