Evaluate `lim_(x to 3) (x^(7) - 2187)/(x - 81)`

Explanation

We have, `underset(x to 3)(lim) (x^(7) - 2187)/(x^(4) - 81) = underset(x to 3)(lim) ((x^(7)) - (3)^(7))/((x^(4))-(3)^(4))`
`= underset(x to 3)(lim) (x^(7)-x^(7))/(x^(4) - 3^(4)) (x - 3)/(x - 3) = underset(x to 3)(lim) (x^(7) - 3^(7))/(((x - 3)/(x^(4) - 3^(4)))/(x - 3))`
`= underset(x to 3)(lim) (x^(7) - 3^(7))/(x - 3) // underset(x to 3)(lim) (x^(4) - 3^(4))/(x - 3) [:' underset(x to a)(lim) (f(x))/(g(x)) = (underset(x to a)(lim) f(x))/(underset(x to a)(lim) g(x))]`
`= (3)^(7-1) // 4(3)^(4 -1) = 7.3^(6) // 4.3^(3) = (7)/(4) 3^(3) = (189)/(4)`