If `f(x)=(x^(2)-3x+2)/(x^(2)-7x+12),` then which of the following limits exists?
`(i)lim_(xtooo) sin^(-1)f(x)" "(ii)lim_(xtooo) cos^(-1)f(x)`

Explanation

`underset(xtooo)lim(x^(2)-3x+2)/(x^(2)-7x+12)`
`underset(xtooo)lim(1-(3)/(x)+(2)/(x^(2)))/(1-(7)/(x)+(12)/(x^(2)))`
`=underset(xtooo)lim(1-(3)/(x))/(1-(7)/(x))" "`(Ignoring division with higher power)
`=1^(+)" "("as "3//xlt7//x,"when "xlt0)`
So, `underset(xtooo)limcos^(-1)f(x)`does not exist as `cos^(-1)x` is defined for `xin[1-,1].`
Also`" "underset(xtooo)lim(1-(3)/(x)+(2)/(x^(2)))/(1-(7)/(x)+(12)/(x^(2)))=underset(xtooo)lim(1-(3)/(x))/(1-(7
)/(x))`
`=1-" "("as "3//xlt7//x,"when "xlt0)`
So, `underset(xtooo)limsin-1f(x)` exists.