Essentials of Calculus: find the limit problem

[Nawz]

Homework Statement

SYNTHESIS

Find limit as x approaches 3 of : (x^3-27)/x-3

Homework Equations

The Attempt at a Solution

I typed it into my calculator and got x^3 - (27)/(x-3) where x^3 wasn't divided by x-3. I really do not know where to start. I know the answer is 27 from the back of the book.

I was thinking it was something like x^3- (3)^3/ (x-3) and then you can cancel out one of the x-3 but I still cannot get 27 and if you plugin 3 into the original equation you get 0.

 

[hgfalling]

Try factoring x³-27

 

[HallsofIvy]

You need to know that $(a- b)(a^2- ab+ b^2)= a^3- b^3$.

 

[sponsoredwalk]

Or, if you ever forget that a³ - b³ = (whatever it actually equals )
It looks like something that can be factored.

Set x³ - 27 equal to zero.

x³ - 27 = 0

Find an x value that makes x³ - 27 = 0 hold, i.e. x = 3

(3)³ - 27 = 0

Then, because x = 3 is an answer, x - 3 = 0 is a factor so divide x - 3 into x³ - 27

______x²_+_..._______​

x - 3 |x³ + 0x² + 0x - 27

x³ - 3x²​

...​

Keep going & you'll factor it & get your answer

 

[hunt_mat]

Or failing that L'Hopital's rule...

Mat

 

[Bohrok]

Or use a substitution with u = x - 3, no factoring cubes or l'Hôpital's rule.