Let `f(x)={{:(x+1", "xlgt0),(2-x", "xle0):}"and"g(x)={{:(x+3", "xlt1),(x^(2)-2x-2", "1lexlt2),(x-5", "xge2):}`
Find the LHL and RHL of g(f(x)) at x=0 and, hence, find `lim_(xto0) g(f(x)).`

Explanation

`xrarr0^(-)impliesf(x)rarrf(0^(-))=2^(+)" "("using" f(x)=2-x)`
or `" "underset(xto0^(-))limg(f(x))=g(2^(+)-3" "("using" g(x)=x-5`)
Also, `xto0^(+)impliesf(x)torarrf(0^(+))=1^(+)" "("using" f(x)=x+1)`
or `" "underset(xto0^(+))limg(f(x))=g(1^(+))=-3" "("using" g(x)=x^(2)-2x-2)`
Hence, `underset(xto0)limg(f(x))` exists and is equal to -3. Therefore,
`underset(xto0)limg(f(x))=-3`