limit as h approaches 0
cubed root of 8+h -2
divided by h
the cubed root sign is under 8+h and not -2
A. 1/12
B.1/4
C.root 2 over 2
D.root 2
E.2
Answer is A but can someone explain
Recall that (a^3-b^3) = (a-b)(a^2+ab+b^2)
Now let a^3 = 8+h and b^3 = 8 and you have
(8+h)-8 = (∛(8+h)-∛8)(∛(8+h)^2 + ∛((8+h)(8)) + ∛8^2)
= (∛(8+h)-2)(∛(8+h)^2+2∛(8+h)+4)
Thus,
(∛(8+h)-2)/h =
(∛(8+h)-2)(∛(8+h)^2+2∛(8+h)+4)
------------------------------------------------
h(∛(8+h)^2+2∛(8+h)+4)
(8+h)-8
------------------------------------------------
h(∛(8+h)^2+2∛(8+h)+4)
= h/(h(∛(8+h)^2+2∛(8+h)+4))
= 1/(∛(8+h)^2+2∛(8+h)+4)
Now let h->0 and you get
1/(∛(8)^2+2∛(8)+4) =1/(4+4+4) = 1/12
We didnt learn that way is there another way to do it
To find the limit as h approaches 0 of the given expression, you can use the concept of limits and algebraic manipulation. Here's a step-by-step explanation:
Step 1: Write the given expression.
cubed root of (8 + h) - 2 / h
Step 2: Simplify the expression inside the cubed root.
cubed root of (8 + h) - 2 / h = [cubed root of (8 + h) - cubed root of 2] / h
Step 3: Apply the cubed root property.
cubed root of (8 + h) - 2 = [(cubed root of (8 + h))³ - 2³] / [(cubed root of (8 + h))² + (cubed root of (8 + h)) + 2]
This step is necessary because when the numerator and denominator involve cubed roots, we can't directly use algebraic transformations. Therefore, we need to use the cubed root property to express the expression without a cubed root.
Step 4: Expand the numerator using the difference of cubes formula.
[(cubed root of (8 + h))³ - 2³] = (cubed root of (8 + h) - 2)([(cubed root of (8 + h))² + (cubed root of (8 + h)) + 2])
Step 5: Simplify the expression.
[cubed root of (8 + h) - 2] / [(cubed root of (8 + h))² + (cubed root of (8 + h)) + 2] = 1 / [(cubed root of (8 + h))² + (cubed root of (8 + h)) + 2]
Step 6: Take the limit as h approaches 0.
lim(h→0) 1 / [(cubed root of (8 + h))² + (cubed root of (8 + h)) + 2] = 1 / [(cubed root of (8))^2 + (cubed root of (8)) + 2]
Step 7: Evaluate the expression.
1 / [(cubed root of (8))^2 + (cubed root of (8)) + 2] = 1 / (2 + 2 + 2) = 1 / 6 = 1/12
Therefore, the limit as h approaches 0 of the given expression is 1/12. Hence, the correct answer is option A.