Limits

[0h_Lola]

Hi, my name's Lola. I haven't had a math course since I graduated High School in 2007 and that was Algebra 3. My college put me in Calc 1 without ever having a Pre-Calc course. I'm struggling. I'm stuck on 2 problems for my homework involving limits. I went to tutoring at my school today but the tutor had to leave early so I didn't make it through. Anyway, I read through that "show all the work you have" but I really have no idea how to even start these problems.

The first is: lim (2(a+h)^2-2a^2)/h; as x approaches 0

I know that I cannot use substitution, because you cannot divide by zero. I tried factoring but that just became a mess and I'm sure I didn't do that right.


And the 2nd:

lim (sqrt x - sqrt a)/x-a; as x approaches a


I don't even know where to get started with this one. I thought I had to get rid of the radicals but I just can't seem to.



I know the answers to these problems, they are in the back of my book. I just don't know how to get them. Any help at all would be appreciated.
Thanks
Lola

 

[tkhunny]

The first is: lim (2(a+h)^2-2a^2)/h; as x approaches 0

I know that I cannot use substitution, because you cannot divide by zero. I tried factoring but that just became a mess and I'm sure I didn't do that right.

Do not use substitution. You have seen why this is so. That division by zero is a bad idea.

Use algebra! \(\displaystyle 2(a+h)^{2}-2a^{2} = 2(a^{2}+2ah+h^{2})-2a^{2}\)

\(\displaystyle = 2a^{2}+4ah+2h^{2}-2a^{2}\)

\(\displaystyle = 4ah+2h^{2}\)

This gives \(\displaystyle \frac{4ah + 2h^2 }{h} = 4a + 2h\) The limit is much easier now.

 

[0h_Lola]

Thank you!