Let f,g and h be the real valued functions defined on R as f(x)= array|c|c| (x)/(|x|), & x…

Question

Let $f,g$ and $h$ be the real valued functions defined on $R$ as

$f(x)= \begin{array}{|c|c|} \frac{x}{|x|}, & x
e 0 \ 1, & x=0 \end{array}$

$g(x)= \begin{array}{|c|c|} \frac{\sin (x+1)}{(x+1)}, & x
e -1 \ 1, & x=-1 \end{array}$

and $h(x)=2[x]-f(x)$, where $[x]$ is the greatest integer $\le x$.
Then the

value of $\underset{x	o 1}{lim}g(h(x-1))$ is :

Accepted Answer

$f(x)=sgn(x)$

$h(x)=2[x]-sgn(x)$

If $x	o {1}^{+}$ then $h(x-1)=2[x]-2-sgn(x-1)$

$=0-1=-1$

\& if $x	o {1}^{-}$ then $h(x-1)=2[x]-2-sgn(x-1)$

$=-2+1=-1$

$	herefore $ $\underset{x	o {1}^{+}}{lim}g(h(x-1))=\underset{x	o {1}^{+}}{lim}\frac{\sin (h(x+1)+1)}{h(x-1)+1}=1$

$\underset{x	o {1}^{-}}{lim}g(h(x-1))=\underset{x	o {1}^{-}}{lim}\frac{\sin (h(x-1)+1)}{h(x-1)+1}=1$

Solutions · 1