Limits

lim(x→∞) (√x⁴+ax³+3x²+bx+2 - √x⁴+2x³-cx²+3x-d)(√x⁴+ax³+3x²+bx+2 + √x⁴+2x³-cx²+3x-d)/√x⁴+ax³+3x²+bx+2 + √x⁴+2x³-cx²+3x-d)

= lim(x→∞) [(a-2)x³ + (3+c)x² + (b-3)x + 2+d]/√x⁴+ax³+3x²+bx+2 + √x⁴+2x³-cx²+3x-d)

now as lim exists and equal to 4 and the highest deg of numerator = 3 while of denominator is 2 ... So, a-2 = 0 ==> a=2

(3+c)/2 = 4 ==> c = 5

now u have to calculate valus of b and d such that √x⁴+ax³+3x²+bx+2 and √x⁴+2x³-cx²+3x-d are defined

after that

(3+c)x²+(b-3)x+(2+d)/ denominator
dividing numerator and denominator by x² we get,

(3+c)+(b-3)/x + (2+d)/x² /denominator

3+c/2=4

c=5
a=2

b and d are any nos

thanks