Limit, Continuity and Differentiability - Medium

Let

f

g

and

h

be real valued functions defined on

\mathbb{R}

f(x) = \begin{cases} x, & x \neq 0 \\ 1, & x = 0 \end{cases}

g(x) = \begin{cases} \displaystyle \frac{\sin(x+1)}{x+1}, & x \neq -1 \\ 1, & x = -1 \end{cases}

h(x) = 2[x] f(x)

where

[x]

is the greatest integer

< x

. Then the value of

\displaystyle \lim_{x \to 1^+} g(h(x-1))