Evaluate: `lim_(n to oo)((1^2+2^2+3^3+...+n^2)(1^3+2^3+3^3+...+n^3))/(1^6+2^6+3^6+...+n^6)`

Explanation

The given limit is
`lim_(ntooo) (sum_(r=1)^(n)r^(2)xxsum_(r=1)^(n)r^(3))/(sum_(r=1)^(n)r^(6))=lim_(tooo)(1/n)(sum_(r=1)^(n)(r/n)^(2)xx1/nsum_(r=1)^(n)(r/n)^(3))/(1/nsum_(r=1)^(n)(r/n)^(6))`
`=int_(0)^(1)x^(2)dxint_(0)^(1)(x^(3)dx)/(int_(0)^(1)x^(6)dx)=([(x^(3))/3]_(0)^(1)[(x^(4))/4]_(0)^(1))/([(x^(7))/7]_(0)^(1))=(1/3xx1/4)/(1/7)=7/12`