Determine the one sided limit of f(x) s x approaches -2 from the right side.
F(X) = (x^2-2x+4)/(x^3+8).
I do not know where to begin. Should I plug in the one side limit?
x²+4x+4=(x+2)(x+2)
x³+8=(x+2)(x²-2x+4)
(x²+4x+4)/(x³+8)=
(x+2)(x+2)/(x+2)(x²-2x+4)
(x+2)/(x²-2x+4)
Lim((x²+4x+4)/(x³+8)) =
Lim((x+2)/(x²-2x+4))=
(-2+2)/((-2)²-2(-2)+4)=0/4=0
Hence Lim((x²+4x+4)/(x³+8)) as x approaches - 2 from the right =0
F(-2) = (4+4+4)/(-8+8)
= 4/0
So, it will be infinity, but plus or minus?
since F(x) = ((x-1)^2 + 3)/(x^3+8), if x is slightly more than -2, the numerator is positive and the denominator is positive, so the limit is plus infinity.
To determine the one-sided limit of a function as x approaches a specific value, you cannot simply plug in that value. Here's how you can find the one-sided limit of f(x) as x approaches -2 from the right side for the given function:
1. Begin by evaluating the function f(x) for values of x that are slightly greater than -2. This will give you an idea of the behavior of the function as x approaches -2 from the right side.
2. Substitute x = -2 + δ into the function, where δ is a small positive number. This represents values of x that are slightly greater than -2.
f(-2 + δ) = ((-2 + δ)^2 - 2(-2 + δ) + 4) / ((-2 + δ)^3 + 8)
3. Simplify the expression:
f(-2 + δ) = (4 - 4δ + δ^2 + 4 + 4δ + 4) / (-8 - 12δ - 6δ^2 - δ^3 + 8)
= (δ^2 + 12) / (-δ^3 - 6δ^2 - 12δ)
4. As δ approaches 0, which represents x approaching -2 from the right side, you can determine the limit of the function:
lim(δ→0+) (δ^2 + 12) / (-δ^3 - 6δ^2 - 12δ)
5. To find this limit, you can factor out the highest power of δ in the denominator, which is δ^3:
lim(δ→0+) (-δ^3 * (-(1/δ^3) * (δ^2 + 12))) / (-δ^3 * (1 + (6/δ) + (12/δ^2)))
6. Cancel out the common factors of δ^3:
lim(δ→0+) (-(δ^2 + 12)) / (1 + (6/δ) + (12/δ^2))
7. Now, as δ approaches 0, the terms containing δ in the denominator go to infinity, so you have:
lim(δ→0+) (-δ^2 - 12) / ∞
8. Since you have a finite value (-δ^2 - 12) divided by infinity, the resulting limit will be 0:
lim(δ→0+) (-δ^2 - 12) / ∞ = 0
Therefore, the one-sided limit of f(x) as x approaches -2 from the right side is 0.