Let f, g and h be the real valued functions defined on R as f(x)=… - Vidyadip

Question

Let $f, g$ and $h$ be the real valued functions defined on $R$ as $f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0\end{array}\right.$,

$g(x)=\left\{\begin{array}{cl}\frac{\sin (x+1)}{(x+1)}, & x \neq-1 \\1, & x=-1\end{array} \text { and } h(x)=2[x]-f(x),\right.$

where $[x]$ is the greatest integer $\leq x$. Then the value of $\lim _{x \rightarrow 1} g(h(x-1))$ is

✓

Answer

a
$LHL =\lim _{ k \rightarrow 0} g ( h (- k )) , k > 0$

$=\lim _{ k \rightarrow 0} g (-2+1) \because f ( x )=-1 \forall x < 0$

$= g (-1)=1$

$RHL =\lim _{ k \rightarrow 0} g ( h ( k )) , k > 0$

$=\lim _{ k \rightarrow 0} g (-1) , \because f ( x )=1, \forall x > 0$

$=1$

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