Use algebra to evaluate the limit of lim h->0 (2-h)^3-8/h.

Is the answer -12

Yes, -12. I cheated and graphed it.

8/h -> infinity as h -> 0, so I don't see how the above expression can tend to anything other than minus infinity.

To evaluate the limit of lim h->0 (2-h)^3-8/h using algebra, we can simplify the expression first, and then substitute h = 0 into the simplified expression. Let's go step by step.

Step 1: Simplify the expression.
We can expand the cubic term (2-h)^3 using the binomial expansion. The binomial expansion of (a-b)^3 is a^3 - 3a^2b + 3ab^2 - b^3. Applying this to (2-h)^3:
(2-h)^3 = 2^3 - 3(2^2)(-h) + 3(2)(-h)^2 - (-h)^3
= 8 - 12h + 6h^2 - h^3

Now we have the expression (8 - 12h + 6h^2 - h^3) - 8/h.

Step 2: Simplify further.
Combine like terms:
8 - 12h + 6h^2 - h^3 - 8/h
= (8 - 8) - 12h + 6h^2 - h^3/h
= - 12h + 6h^2 - h^3/h

Step 3: Divide by h.
To divide the expression by h, we can factor out an h from each term:
-12h/h + 6h^2/h - h^3/h
= -12 + 6h - h^2

Now we have the expression -12 + 6h - h^2.

Step 4: Substitute h = 0.
To find the limit, substitute h = 0 into the expression:
-12 + 6(0) - (0)^2
= -12 + 0 - 0
= -12

Therefore, the limit of lim h->0 (2-h)^3-8/h is indeed -12.