Which of the following limits exists finitely?

Options

A. `underset(xrarr0^(+))(lim)(x)^(log_(e)x)`
B. `underset(xrarr1^(+))(lim)(x^(2)-9-sqrt(x^(2)-6x+6))/(|x-1|-2)`
C. `underset(xrarr1^(+))(lim)([x])^((1)/(x-1))=` (where [.] denotes the greatest integer function)
D. none of these

Answer

C

Explanation

(a) `underset(xrarr0^(+))(lim)(x)^(log_(e)x=(0^(+))^(-oo)=oo`
So, limits does not exist finitely.
(b) `underset(xrarr3)(lim)(x^(2)-9-sqrt(x^(2)-6x+9))/(|x-1|-2)`
`=underset(xrarr3)(lim)(x^(2)-9-|x-4|)/(x-3)`
Now,
`f(3^(+))=underset(xrarr3^(+))(lim)((x^(2)-9)-(x-3))/((x-3))=underset(xrarr3^(+))(lim)((x+3)+1)=5`
Also,
`f(3^(-))=underset(xrarr3^(-))(lim)((x^(2)-9)+(x-3))/((x-3))=underset(xrarr3(-))(lim)((x+3)+1)=7`
So, limit does not exist.
(c) `underset(xrarr3^(-))(lim)([x])^((1)/(x-1))underset(xrarr1^(+))(lim)^((1)/(x-1))=1`