Given a real-valued function `f` such that `f(x)={(tan^2{x})/((x^2-[x]^2) sqrt({x}cot{x}),for x<0,fx >0` Where `[x]` is the integral part and `{x}` is the fractional part of `x ,` then `("lim")_(xvec0^+)f(x)=1` , `("lim")_(xvec0^-)f(x)=cot1`, `cot^(-1)(("lim")_(xvec0^-)f(x))^2=1`, `tan^(-1)(("lim")_(xvec0^+)f(x))=pi/4`

Options

A. `p_(1)" ln "a_(1)+p_(2)" ln "a_(2)+...+p_(n)" ln "a_(n)`
B. `a_(1)^(p_(1))+a_(2)^(p_(2))+...+a_(n)^(p_(n))`
C. `a_(1)^(p_(1)).a_(2)^(p_(2))...a_(n)^(p_(n))`
D. `sum_(r=1)^(n)a_(r)p_(r)`

Answer

A, B, C, D

Explanation

We have `f(x)=underset(xto0^(+))lim(tan^(2){x})/((x^(2)-[x]^(2)))=underset(xto0^(+))lim(tan^(2)x)/(x^(2))=1" "(1)`
`(becausexto0^(+),[x]=0implies{x}=x)`
Also, `underset(xto0^(-))limf(x)=underset(xto0^(-))limsqrt({x}cot{x})=sqrt(cot1)" "(2)`
`(becausexto0^(-),[x]=-1implies{x}=x+1implies{x}to1)`
Also,`cot^(-1)(underset(xto0^(-))limf(x))^(2)=cot^(-1)(cot1)=1.`
Also, `tan^(-1)(underset(xto0^(+))limf(x))=tan^(-1)1=(pi)/(4)`