Which of the following true ({.} denotes the fractional part of the function)? `("lim")_(xvecoo)((log)_e x)/({x})=oo` (b) `("lim")_(xvec2^+)x/(x^2-x-2)=oo` `("lim")_(xvec1^-)x/(x^2-x-2)=-oo` (d) `("lim")_(xvecoo)((log)_(0.5) x)/({x})=oo`

Options

A. always 1
B. always -1
C. `(-1)^(n-m+1)`
D. `(-1)^(n-m)`

Answer

A, B, C

Explanation

`underset(xtooo)lim(log_(e)x)/({x})=("positive infinity")/("a value between 0 and 1")=oo`
`underset(xto2^(+))lim(x)/(x^(2)-x-2)=underset

(xto2^(+))lim(x)/((x-2)(x+1))=underset(hto0)lim(2+h)/(h(3+h))=oo`
`underset(hto-1)lim(x)/(x^(2)-x-2)=-underset(hto-1)lim(x)/((x-2)(x+1))`
`=underset(hto0)lim(-1-h)/((-3-h)(-h))=-underset(hto0)lim(1+h)/((3+h)(h))=-oo`