To find the limit of the function as x approaches 2, we can first simplify the function by factoring out an x^2 from the numerator and denominator:
x3−3x2+4x^3 - 3x^2 + 4x3−3x2+4 / x4−4x2x^4 - 4x^2x4−4x2 = x^2 x−3+4/x2x - 3 + 4/x^2x−3+4/x2 / x^2 x2−4x^2 - 4x2−4
Now we can cancel out the common x^2 terms in the numerator and denominator:
= x−3+4/x2x - 3 + 4/x^2x−3+4/x2 / x2−4x^2 - 4x2−4
Next, substitute x = 2 into the simplified function:
2−3+4/222 - 3 + 4/2^22−3+4/22 / 22−42^2 - 422−4 = 2−3+4/42 - 3 + 4/42−3+4/4 / 4−44 - 44−4 = −1+1-1 + 1−1+1 / 0= 0 / 0
Since we obtained an indeterminate form of 0 / 0, we can further simplify the function by factoring out the common x−2x - 2x−2 term in the numerator:
= (x−2)−1+4/x2(x - 2) - 1 + 4/x^2(x−2)−1+4/x2 / x+2x + 2x+2x−2x - 2x−2
Now, substitute x = 2 into the simplified function:
= (2−2)−1+4/22(2 - 2) - 1 + 4/2^2(2−2)−1+4/22 / 2+22 + 22+22−22 - 22−2 = 0−1+10 - 1 + 10−1+1 / 444 = 0
Therefore, the limit of the function as x approaches 2 is 0.
To find the limit of the function as x approaches 2, we can first simplify the function by factoring out an x^2 from the numerator and denominator:
x3−3x2+4x^3 - 3x^2 + 4x3−3x2+4 / x4−4x2x^4 - 4x^2x4−4x2 = x^2 x−3+4/x2x - 3 + 4/x^2x−3+4/x2 / x^2 x2−4x^2 - 4x2−4
Now we can cancel out the common x^2 terms in the numerator and denominator:
= x−3+4/x2x - 3 + 4/x^2x−3+4/x2 / x2−4x^2 - 4x2−4
Next, substitute x = 2 into the simplified function:
2−3+4/222 - 3 + 4/2^22−3+4/22 / 22−42^2 - 422−4 = 2−3+4/42 - 3 + 4/42−3+4/4 / 4−44 - 44−4 = −1+1-1 + 1−1+1 / 0
= 0 / 0
Since we obtained an indeterminate form of 0 / 0, we can further simplify the function by factoring out the common x−2x - 2x−2 term in the numerator:
= (x−2)−1+4/x2(x - 2) - 1 + 4/x^2(x−2)−1+4/x2 / x+2x + 2x+2x−2x - 2x−2
Now, substitute x = 2 into the simplified function:
= (2−2)−1+4/22(2 - 2) - 1 + 4/2^2(2−2)−1+4/22 / 2+22 + 22+22−22 - 22−2 = 0−1+10 - 1 + 10−1+1 / 444 = 0
Therefore, the limit of the function as x approaches 2 is 0.