Use algebra to evaluate the limit of lim h->0 (2-h)^(3)-8/h.
Is the answer -12
correct.
Why is the answer -12, please? I'm seeing the difference between two terms, the first of which approaches (2^3)=8 as h tends to zero, and the second of which approaches infinity as h tends to zero.
To evaluate the limit of the expression lim h->0 (2-h)^(3)-8/h using algebra, we need to simplify the expression and then substitute h=0 into the simplified expression.
Step 1: Simplify the expression:
First, expand the binomial (2-h)^(3) using the binomial theorem or by multiplying it out:
(2-h)^(3) = (2-h)(2-h)(2-h) = (4 - 4h + h^2)(2-h) = 8 - 8h + 2h^2 - 4h + 4h^2 - h^3 = 8 - 12h + 6h^2 - h^3
Step 2: Substitute h=0:
Now, substitute h=0 into the simplified expression:
lim h->0 (8 - 12h + 6h^2 - h^3) - 8/h
Substituting h=0,
lim h->0 (8 - 12(0) + 6(0)^2 - (0)^3) - 8/0
Simplifying,
lim h->0 (8 - 0 + 0) - undefined
Since the denominator is zero, the expression is undefined. Therefore, the limit does not exist.
So, the answer is not -12.