Dengan menyederhanakan lebih dahulu (menyamakan penyebut),

a. ∞, b. -1, c. -1, d. -1/24

Pembahasan

Untuk menghitung limit fungsi tersebut, kita perlu menyederhanakan ekspresi dengan menyamakan penyebutnya terlebih dahulu.

a. lim x -> 0 (1/(x^2-x) + 1/x)
= lim x -> 0 (1/(x(x-1)) + x/(x(x-1)))
= lim x -> 0 ((1+x)/(x(x-1)))
Substitusikan x=0: (1+0)/(0(0-1)) = 1/0. Limitnya adalah ∞.

b. lim x -> 0 (2/(x^2-1) - 1/(x-1))
= lim x -> 0 (2/((x-1)(x+1)) - (x+1)/((x-1)(x+1)))
= lim x -> 0 ((2 - (x+1))/((x-1)(x+1)))
= lim x -> 0 ((1-x)/((x-1)(x+1)))
= lim x -> 0 (-1/(x+1))
Substitusikan x=0: -1/(0+1) = -1.

c. lim x -> 1 (1/(1-x) - 3/(1-x^3))
= lim x -> 1 (1/(1-x) - 3/((1-x)(1+x+x^2)))
= lim x -> 1 ((1+x+x^2 - 3)/((1-x)(1+x+x^2)))
= lim x -> 1 ((x^2+x-2)/((1-x)(1+x+x^2)))
= lim x -> 1 (((x+2)(x-1))/((1-x)(1+x+x^2)))
= lim x -> 1 (-(x+2)/(1+x+x^2))
Substitusikan x=1: -(1+2)/(1+1+1^2) = -3/3 = -1.

d. lim x -> 2 (2/(x^2-4) - 3/(x^2+2x-8))
= lim x -> 2 (2/((x-2)(x+2)) - 3/((x-2)(x+4)))
= lim x -> 2 ( (2(x+4) - 3(x+2)) / ((x-2)(x+2)(x+4)) )
= lim x -> 2 ( (2x+8 - 3x-6) / ((x-2)(x+2)(x+4)) )
= lim x -> 2 ( (-x+2) / ((x-2)(x+2)(x+4)) )
= lim x -> 2 ( -(x-2) / ((x-2)(x+2)(x+4)) )
= lim x -> 2 ( -1 / ((x+2)(x+4)) )
Substitusikan x=2: -1 / ((2+2)(2+4)) = -1 / (4*6) = -1/24.