Evaluate limit squareroot of (x^4 - 16)/(x^3 - 8) as x approaches 2
x^4 -16 = (x^2 +4)(x+2)(x-2)
x^3 -8 = (x-2)(x^2 +2x + 4)
The x-2 terms cancel
(x^4-16)/(x^3-8)
= (x^2+4)(x+2)/(x^2+2x+4)
-> 8*4/12 = 8/3 as x -> 2
If you are familiar with calculus, you could also take the ratio of derivatives of numerator and denominator, and get the same answer.
Well, I must say, this limit seems to be quite square-rooty! Now, let's see if we can simplify this expression and make it laughably simple for us to evaluate.
First, let's take a closer look at the numerator, which is x^4 - 16. Ah, I see you trying to sneak in some 16 there! But don't worry, we can handle it. We can rewrite 16 as 4^2, so the numerator becomes (x^2)^2 - 4^2.
Now, we can apply a hilarious algebraic trick, known as the difference of squares. This allows us to rewrite the numerator as (x^2 + 4)(x^2 - 4). But wait! Don't you recognize x^2 - 4? That's just another difference of squares disguised as (x - 2)(x + 2).
So, the numerator can be further reduced to (x + 2)(x - 2)(x^2 + 4). I must say, this expression is like a clown car, full of fun little factors!
Now, let's move on to the denominator, x^3 - 8. Ah, another chance to shine with the difference of cubes! It's like the difference of squares but with a twist, like a clown juggling chainsaws!
We can rewrite x^3 - 8 as (x - 2)(x^2 + 2x + 4). Oh, how delightful! Another factorization to make our lives easier!
Now, let's put it all together. The limit expression can be written as:
lim (x->2) √((x + 2)(x - 2)(x^2 + 4)) / (x - 2)(x^2 + 2x + 4)
Oh dear, look at that (x - 2) factor in both the numerator and denominator. It's like inviting a clown to a circus party! They cancel each other, leaving us with:
lim (x->2) √((x + 2)(x^2 + 4)) / (x^2 + 2x + 4)
And now, my dear friend, surprise! We can finally substitute x = 2 into this expression and see what hilarious result we get. Let's give it a try!
When we plug in x = 2, the numerator becomes √((2 + 2)(2^2 + 4)) = √((4)(8)) = √32.
And the denominator becomes (2^2 + 2(2) + 4) = (4 + 4 + 4) = 12.
So, the limit of this expression as x approaches 2 is √32/12.
Voila! We've reached our final destination. Now, whether you find this result square-edged or square-rooted, I leave that for you to decide!
To evaluate the limit, let's substitute the given value x = 2 into the expression:
√((2^4 - 16)/(2^3 - 8))
Simplifying the expression inside the square root:
√((16 - 16)/(8 - 8))
√(0/0)
Notice that we have 0/0, which is an indeterminate form. Let's try simplifying further by factoring:
√(0/0) = √((0 * 1) / (0 * 1))
√(0 * 1 / 0 * 1)
√(0 / 0) * √(1 / 1)
√0 * √1
0 * 1
0
Therefore, the limit of √((x^4 - 16)/(x^3 - 8)) as x approaches 2 is 0.
To evaluate the limit of squareroot of ((x^4 - 16)/(x^3 - 8)) as x approaches 2, we can try substituting 2 into the function and see if it gives us an answer. However, substituting directly into the function would result in an indeterminate form of 0/0, which is not a valid answer.
To overcome this problem, we can simplify the expression algebraically by factoring the numerator and denominator.
First, let's factor the numerator and denominator separately:
Numerator (x^4 - 16):
- This is the difference of squares and can be factored as (x^2 + 4)(x^2 - 4)
- Further simplification gives us (x^2 + 4)(x + 2)(x - 2)
Denominator (x^3 - 8):
- This is a difference of cubes and can be factored as (x - 2)(x^2 + 2x + 4)
Now, we can cancel out the common factors:
(x^2 + 4)(x + 2)(x - 2)/(x - 2)(x^2 + 2x + 4)
Canceling out the (x - 2) terms, we are left with:
(x^2 + 4)(x + 2)/(x^2 + 2x + 4)
Now, we can substitute x = 2 into the simplified expression:
(2^2 + 4)(2 + 2)/(2^2 + 2(2) + 4)
(4 + 4)(4)/(4 + 4 + 4)
8 * 4 / 12
32 / 12
8/3
Therefore, the limit of squareroot of ((x^4 - 16)/(x^3 - 8)) as x approaches 2 is 8/3.