Evaluate `lim_(xto-3) (x^(3)+27)/(x^(5)+243)`

Explanation

Given, `underset(xto-3)"lim"(x^(3)+27)/(x^(5)+243)=underset(xto-3)"lim"((x^(3)+27)/(x+3))/((x^(5)+243)/(x+3))`
`=underset(xto-3)"lim"((x^(3)-(-3)^(2))/(x-(-3)))/((x^(5)-(-3)^(2))/(x-(-3)))=(underset(xto-3)"lim"(x^(3)-(-3)^(2))/(x-(-3)))/(underset(xto-5)"lim"(x^(3)-(-3)^(5))/(x-(-3)))` `[therefore underset(xtoa)"lim"(f(x))/(g(x))=(underset(xtoa)"lim"f(x))/(underset(xtoa)"lim"g(x))]`
`=(3(-3)^(3-1))/(5(-3)^(5-1))= (3(-3)^(2))/(5(-3)^(4))` `[therefore underset(xtoa)"lim"(x^(n)-a^(n))/(x-a)=na^(n-1)]`
`=(3)/(5(-3)^(2))=3/45=1/15`