Evaluate the limit as h approaches zero of
(the cubed root of 8-h)-2/h.
Can also be written ((8-h)^1/3)-2)/h
let (8-h)^(1/3) = x
cube both sides
8-h = x^3
h = 8-x^3
as h--->0 , x ----> 2
so we write our limit as
lim (x - 2)/(8 - x^3) as x --> 2
= lim (x-2)/((x-2)(x^2 + 2x + 4) as x--2
= lim 1/(x^2 + 2x + 4)
= 1/(4+4+4) = 1/12
Ah, the limit as h approaches zero, a classic problem in the world of calculus. But fear not, because I'm here to solve it with a touch of clown humor!
Alright, let's tackle this. We've got the expression (((8 - h)^(1/3)) - 2) / h, or in a more rooty way, (cuberoot(8 - h) - 2) / h. Now, let's find out what happens when h gets really close to zero.
Well, think about it: when h is super close to zero, the term (8 - h)^(1/3) is essentially like taking the cubed root of 8. And hey, what's the cubed root of 8? It's 2! So we can simplify the expression as (2 - 2) / h.
Now, (2 - 2) is zero, and we're dividing that by h. No matter what, dividing zero by anything will always give you... you guessed it, zero! So the answer to this limit problem is 0.
And that, my friend, is the math equivalent of a clown slipping on a banana peel. Enjoy!
To evaluate the limit as h approaches zero of ((8-h)^(1/3) - 2)/h, we can use a combination of algebraic manipulations and limit properties. Let's work through the steps:
Step 1: Simplify the expression.
Rewriting ((8-h)^(1/3) - 2)/h, we can use the property of subtracting fractions to rewrite it as ((8-h)^(1/3))/h - 2/h.
Step 2: Rationalize the numerator.
To rationalize the numerator, we multiply the expression by the conjugate of the numerator, (∛(8-h) + 2)/(∛(8-h) + 2). Multiplying the numerator and denominator by this conjugate, we get:
[((8-h)^(1/3))/h - 2/h] * [(∛(8-h) + 2)/(∛(8-h) + 2)].
Step 3: Simplify the expression.
Multiplying the numerators and denominators, we have:
[((8-h)^(1/3)) * (∛(8-h) + 2)] / [h * (∛(8-h) + 2)].
Step 4: Cancel out like terms.
Since the numerator and denominator both contain (∛(8-h) + 2), we can cancel them out, resulting in:
(8-h)^(1/3) / h.
Step 5: Simplify further.
We can rewrite (8 - h)^(1/3) as [((8 - h)^(1/3)) - 2^(1/3) + 2^(1/3)], using the property of adding and subtracting the same quantity. This gives us:
[((8 - h)^(1/3)) - 2^(1/3) + 2^(1/3)] / h.
Step 6: Apply the limit.
Now, we take the limit as h approaches zero. Plugging in h = 0 into the expression, we have:
[((8 - 0)^(1/3)) - 2^(1/3) + 2^(1/3)] / 0.
Since we have 0 in the denominator, the expression is undefined.
Therefore, the limit as h approaches zero of ((8-h)^(1/3) - 2)/h is undefined.
To evaluate the given limit, we can start by simplifying the expression inside the limit by rationalizing the numerator. Let's break down the steps:
Step 1: Rationalize the numerator.
The expression (the cubed root of 8-h) can be rationalized by multiplying both the numerator and denominator by the conjugate of the numerator, which is (∛(8-h))^2 + (∛(8-h)) * 2 + 2^2. This will eliminate the radical in the numerator.
((8-h)^(1/3) - 2)/h * (∛(8-h))^2 + (∛(8-h)) * 2 + 2^2 / (∛(8-h))^2 + (∛(8-h)) * 2 + 2^2
Simplifying the numerator:
((8-h)^(1/3) - 2) * ((8-h)^(2/3) + 2(8-h)^(1/3) + 4)
(8-h) - 2(8-h)^(1/3) + 16 - 4(8-h)^(1/3) - 2(8-h)^(2/3) + 4(8-h)^(1/3)
8 - h - 2(8-h)^(1/3) + 16 - 4(8-h)^(1/3) - 2(8-h)^(2/3) + 4(8-h)^(1/3)
24 - h - 6(8-h)^(1/3) - 2(8-h)^(2/3)
Step 2: Simplify the denominator.
The denominator is h, and as h approaches zero, the denominator becomes zero.
Now that we have simplified the expression inside the limit, we can substitute h = 0 into the expression:
lim(h→0) (24 - h - 6(8-h)^(1/3) - 2(8-h)^(2/3)) / h
Since the denominator is 0, we cannot directly substitute h = 0. Instead, we can simplify further by dividing each term by h:
lim(h→0) (24/h) - (h/h) - (6(8-h)^(1/3))/h - (2(8-h)^(2/3))/h
Simplifying this expression, we have:
lim(h→0) 24/h - 1 - (6(8-h)^(1/3))/h - (2(8-h)^(2/3))/h
Now, we can substitute h = 0 into the expression:
lim(h→0) 24/0 - 1 - (6(8-0)^(1/3))/0 - (2(8-0)^(2/3))/0
Since the numerator remains finite while the denominator approaches 0, the overall limit becomes undefined.