limit laws and manipulating equations

[trigfun]

Hello,

I'm having trouble figuring out how to approach this problem. is throwing me off. Do I foil? I understand that I can't use limit laws with the equation in this form, I need to manipulate the equation algebraically. I'm just having trouble seeing how. Here's the problem:

Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)



I'd really appreciate some tips on where to start with this.

Thank you!
Jill

 

[MarkFL]

Rather than expanding (which would work), you could also use a difference of cubes factorization:

[MATH](7+h)^3-7^3=(7+h-7)((7+h)^2+(7+h)7+7^2)=h(h^2+21h+147)[/MATH]

 

[trigfun]

Thank you! That all makes sense.

Now I'm not sure what to do with the denominator. As it stands, h would equal zero, so I know I need to manipulate the equation to deal with that. What comes to mind is multiplying by a well-chosen one, but I don't see how I could do that here. Any tips?

Thank you!

 

[Deleted member 4993]

Thank you! That all makes sense.

Now I'm not sure what to do with the denominator. As it stands, h would equal zero, so I know I need to manipulate the equation to deal with that. What comes to mind is multiplying by a well-chosen one, but I don't see how I could do that here. Any tips?

Thank you!

Your

numerator = h * (h^2 + 21h + 147)
denominator = h

What do you get when you divide the numerator by the denominator?

Now take the limit......

 

[trigfun]

Ohh, ok, thank you!

 

[HallsofIvy]

The definition of the derivative is \(\displaystyle \lim_{h\to 0}\frac{f(x+ h)- f(x)}{h}\). The denominator always goes to 0! You had better get used to that! That means that the numerator must also go to 0 so you have the "indeterminate" \(\displaystyle \frac{0}{0}\) and will need to use some more subtle method of evaluating the limit than just setting h equal to 0.