How would I find the limit of (if it exists):
lim x-->3 (x^3-27)
--------
(x^2-9)
It might look better this way...
lim x-->3 (x^3 -27)/(x^2 -9)
it factors
lim x-->3 (x^3 -27)/(x^2 -9) ax x--->3
= lim [(x-3)(x^2 + 3x + 9)]/[(x+3)(x-3)]
= lim (x^2 + 3x + 9)/(x+3)
= (9+9+9)/(3+3)
= 9/2
To find the limit of the expression as x approaches 3, you can start by plugging in the value 3 into the expression and see what you get. However, if you encounter any division by zero or an undefined expression, it indicates the need for further simplification or factoring.
In this case, let's simplify the expression first. Notice that both the numerator and denominator are difference of cubes, which can be factored using the formula:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Using this formula, the numerator (x^3 - 27) can be factored as (x - 3)(x^2 + 3x + 9), and the denominator (x^2 - 9) can be factored as (x - 3)(x + 3).
Now, the expression becomes:
lim x-->3 (x - 3)(x^2 + 3x + 9)
---------------------
(x - 3)(x + 3)
Next, we can cancel out the common factors of (x - 3) from both the numerator and the denominator:
lim x-->3 (x^2 + 3x + 9)
---------------
(x + 3)
Now, we can substitute x=3 into the simplified expression:
(3^2 + 3(3) + 9) / (3 + 3)
Simplifying further:
(9 + 9 + 9) / 6
27 / 6
The result is 4.5.
Therefore, the limit of the given expression as x approaches 3 is 4.5.