Calculus Limits: Verifying the Solution for (x+∆x)^2 - x^2 / x

[kendal12]

I just started calculus and we are doing limits. I was doing fine until I ran across this problem:

lim ( (x+∆x)^2 - x^2 ) / x
∆x approaches 0

I substituted delta x for a and ended up with 2x + a

I don't think this is right for some reason, but I think I did everything right, can someone please verify my answer so that I know I am on the right track? thanks!

 

[courtrigrad]

Just expand it out. (note that $a = \Delta x)$ $(x+a )^{2} - x^{2} = 2ax + a^{2}$. So we have $\frac{2ax + a^{2}}{x}$. Divide through by $x$. What do you notice when you substitute $a = 0$.

 

[kendal12]

I actually messed up when I typed that problem. It should all be over delta x. I'm so sorry! So I divided through by a and got a + 2x. Then substituted o in for a, so I'm left with 2x. Right?

 

[lalbatros]

2x, you are right !
Remember this calculation, since you will learn soon that 2x is the derivative of x².