kill(all)$ e:%e$

assume(n>0);

limit(product(e^(k*x)-1,k,1,n)/x^n,x,0);

limit(0/x^n,x,0);

A:matrix
(
    [e^(2*x+sin(x)), e^(-3*x),e^(a*x)+e^(b*x)+e^(c*x)],
    [e^sin(x),e^(2*x)+4*x,e^(2*x)+e^(4*x)],
    [e^(x^3)+e^(x*sin(x)^2*2*x),e^(2*x^2),e^(x*tan(2*x))],
    [e^sin(x)+e^(-sin(x)),2*e^(3*x)-3*e^(2*x),4*e^(-x)-5*e^(2*x)],

    [diff(e^(1+x*sin(2*x))-e^cos(2*x),x), e^sin(2*x), e^(3*sin(x))],
    [1, 0, 1+x^2],
    [e^(1-x^2), e^(x+1)^2, e^(2-cos(x))],
    [e^cos(2*x), e^sqrt(1+x*sin(2*x)), e^(tan(x)^2+cos(2*x))],

    [e^(x^2+2*x), e^sqrt(1+x^2), 1],
    [e^cos(a*x), e^sin(x)^2, e^cos(2*x)],
    [e^(x^3+3*x^2), e^(1-sqrt(cos(2*x))), 1],
    [e^(1+sin(3*x)^2), diff(e^(1+3*sin(2*x))-e^(2-cos(4*x)),x,2), 1],

    [2^x, a^x+b^x, 6^x],
    [2^(x+1), 4, diff(2^cos(x)-2^x^2,x,2)],
    [3^tan(x), 2^cos(2*x), sqrt(1+2^(x+3))],
    [4^cos(2*x), a^x+a^(-x), 1]
)$

B:matrix
(
    [1, e^(sin(4*x)),3],
    [e^(-sin(x)),1,2],
    [2,cos(4*x),2-cos(2*x)],
    [2,-1,-1],

    [0, e^tan(2*x), e^sin(3*x)],
    [e^x^2*cos(2*x), 0, e^x^2*cos(4*x)],
    [e^cos(4*x), e^(x-1)^2, e^(1+sin(3*x)^2)],
    [e^cos(4*x), e^cos(2*x), e^(1+sin(3*x)^2)],

    [e^(x^2-2*x), e^cos(2*x), e^sin(x)^2*sqrt(cos(x))],
    [e^cos(b*x), e^(1-sqrt(cos(2*x))), e^(cos(x)*sqrt(cos(2*x)))],
    [e^(x^3-3*x^2), e^tan(x)^2, e^sin(x)^2*cos(2*x)*cos(4*x)],
    [e^(cos(x)*cos(3*x)), 0, (1+x^2)*e^x^2],

    [3^x, 2, 2^x],
    [2*e^x, (1+a^x)*(1+b^x), 0],
    [3^(-tan(x)), 2^cos(4*x), sqrt(3^(x+2))],
    [2^(1+cos(4*x)), 2, 2^x^2*cos(2*x)]
)$

C:matrix
(
    [x, x, x], 
    [x, e^(4*x)-x-1,e^(3*x)-1],
    [x^3,x^2,x^2],
    [x^2, x^2, x],

    [diff(sqrt(1+x^2)-1,x), x^3, 1-sqrt(1-x^3)],
    [tan(x)^2, x^n, x^2+1-cos(2*x)],
    [x^2, e^(x+1)-e^(1-x), e^(1+x^2)-e^(1-x^2)],
    [e^x+e^(-x)-2, x^2, x^2],

    [x, x^2, x^2],
    [x^2, x^2, x^2],
    [e^(1+x^2)-e^cos(2*x), x^2, x^2],
    [x^2, diff(x^2,x,2), 1-e^(-x^2)*cos(2*x)],

    [x, a^x-b^x, 4^x-2^x],
    [x, x, diff(x^2,x,2)],
    [x, x^2, x],
    [x^2, x^2, x^2]

)$

f[i,j]:=limit((A[i,j]-B[i,j])/C[i,j],x,0)$

M:genmatrix(f,16,3)$ M;