If `f(x)={{:((x)/(sinx)",",x gt0),(2-x",",xle0):}andg(x)={{:(x+3",",xlt1),(x^(2)-2x-2",",1lexlt2),(x-5",",xge2):}`
Then the value of `lim_(xrarr0) g(f(x))`

Options

A. is `-2`
B. is `-3`
C. is 1
D. does not exist

Answer

B

Explanation

`underset(xrarr0+)(lim)g(f(x))=g(f(0^(+)))=g(1^(+))" "(becauseunderset(xrarr0)(lim)(x)/(sinx)=1^(+))`
`" "=1-2(1)-2=3`
`underset(xrarr0^(-))(lim)g(f(x))=g(f(0^(-)))=g(2^(+))" "(becauseunderset(xrarr0^(-))(lim)(2-x)=2^(+))`
`" "=2-5=-3`
`rArr" "underset(xrarr0)(lim)g(f(x))=-3`