Question

Can anyone please help me in Calculus? I am trying to study my butt off for my first test and I have some problems I don't know how to do like this: http://i.imgur.com/SbfgaE2.jpg

If you have the time, please help me! I'll be eternally grateful!!

Answer 1

\[\lim_{x \rightarrow 4}(\frac{ x ^{3}-64 }{ 4-x })\]
Okay, we have this limit wich is a undefines one, let's try to get rid of that (4-x) so we can find a value for it.

Answer 2

okay :)

Answer 3

i tried factoring but i cant get it. im probably doing it wrong :<

Answer 4

Let's try together:
\[\lim_{x \rightarrow 4}(\frac{ 4^{3}-64 }{ 4-x })\]
Let's use the ruffini division to factor that:

Answer 5

what. so do i just do 4-x/(x^3 - 64)

Answer 6

x^3 - 64 is a difference of cubes. That factors like this:
\[(a ^{3}-b ^{3})=\]
\[(a-b)(a ^{2}+ab + b ^{2})\]

The (a-b) term matches the sign of the original problem.
The ab term is always the opposite sign. Since our original was x^3 - 64, this makes ab positive. The b^2 is always positive. So given a is x and b is 4 (thats the cube root of 64), we can factor it like this:

\[\frac{ (x-4)(x ^{2}+4x+16) }{ 4-x }\]

So as you may notice, the top factor of (x-4) is pretty close to the bottom factor of (4-x). We can get those two factors to match and cancel out by factoring out a negative from the top or bottom either works. So if I take the bottom factor of (4-x) and factor out a negative, i can rewrite it as -(x-4). Now that we have that, we can cancel out the top and bottom and just tack on the negative sign, which leaves us:
\[\lim_{x \rightarrow 4}-(x ^{2}+4x+16)\]

Now thatwe have this, we dont have to worry about an undefined answer, we can just straight away plug in x = 4
\[\lim_{x \rightarrow 4}-((4)^{^{2}}+4(4) + 16)= -(48)\]

Answer 7

Yup, i got that same answer.

Answer 8

thank you!! so the division works too, right? also, is it wise to memorize that difference of cubes formula in calculus?

Answer 9

You wont see it super often admittedly, but it always helps. You can do division too, though, yes. My first instinct is to factor if I can, though.

Answer 10

The result above is just the same thing that I had, excpet I had the negative factored out.