Question

Let f(x)=$\{\begin{array}{cc}x+1,x>0& \\ 2-x,x\le 0& \end{array}$ and g(x)=$\{\begin{array}{cc}x+1,x<1& \\ x^2-2x-2,1\le x<2& \\ x-5,x\ge 2& \end{array}$
Find the LHL and RHLof g(f(x)) at x=0 and, hence, find $\underset{x\to 0}{lim}g\left(f\right(x\left)\right)$.

Answer 1

$f\left(x\right)=\{\begin{array}{cc}x+1& x>0\\ 2-x& x\le 0\end{array}$

$g\left(x\right)=\{\begin{array}{cc}x+1& x<1\\ x^2-2x-2& 1\le x\le 2\\ x-5& x\ge 2\end{array}$

$g\left(f\right(x\left)\right)=\{\begin{array}{cc}f\left(x\right)f& f\left(x\right)<1\\ f(x{)}^2-2f(x)-2& 1\le f\left(x\right)\le 2\\ f\left(x\right)-5& f\left(x\right)\ge 2\end{array}$

${<}_{x\to 0}g\left(f\right(x\left)\right)=(2-x{)}^2-2(2-x)-2$

$=4-4-2$

$=-2$.