Expert: Bobby Soltani - 5/13/2005

Question
Can you please show me all the different ways to do the following problem:
What is the limit of [(x+h)^2 -(x)^2]/h as h approaches 0?
Is there a way to do it with L'hopitals rule or derivitaves?If yes,how?Please explain.
Thank you very much for helping me.

Answer
Hi Jeff,

I don't think you need L'hopitals rule or derivatives.  You only need L'hopitals rule when h would be zero in the denominator.  We can cancel the h in the denominator.  Here's how.

I'm just going to write "lim" to represent limit as h approaches zero.

lim [(x+h)^2 - x^2]/h

lim [x^2 + 2xh + h^2 - x^2]/h

lim [2xh + h^2]/h

lim [2x + h]
now we can plug in zero for h.

= 2x.

Note that this is the derivative for the function f(x) = x^2.  Using the chain rule for f(x) = x^2, we also get f'(x) = 2x.

Let me know if you have any questions.

Bobby