$f(x) = \left\{ {\begin{array}{cccccccccccccccccccc}{\frac{{\sqrt {1 + kx} - \sqrt {1 - kx} }}{x},}&{{\rm{ if }} - 1 \le x < 0}\\{\frac{{2x + 1}}{{x - 1}},}&{{\rm{ if }}0 \le x \le 1}\end{array}} \right.$at $x = 0$}

We have, $f(x) = \left\{ {\begin{array}{cccccccccccccccccccc}{\frac{{\sqrt {1 + kx} - \sqrt {1 - kx} }}{x},}&{{\rm{ if }} - 1 \le x < 0}\\{\frac{{2x + 1}}{{x - 1}},}&{{\rm{ if }}0 \le x \le 1}\end{array}} \right.$at $x = 0$

therefore,${\rm{LHL}} = \mathop {\lim }\limits_{x \to {0^ - }} \frac{{\sqrt {1 + kx} - \sqrt {1 - kx} }}{x}$

$= \mathop {\lim }\limits_{x \to {0^ - }} \left( {\frac{{\sqrt {1 + kx} - \sqrt {1 - kx} }}{x}} \right) \cdot \left( {\frac{{\sqrt {1 + kx} + \sqrt {1 - kx} }}{{\sqrt {1 + kx} + \sqrt {1 - kx} }}} \right)$

$= \mathop {\lim }\limits_{x \to {0^ - }} \frac{{1 + kx - 1 + kx}}{{x[\sqrt {1 + kx} + \sqrt {1 - kx} ]}}$
$= \mathop {\lim }\limits_{x \to {0^ - }} \frac{{2kx}}{{x\sqrt {1 + kx} + \sqrt {1 - kx} }}$

$= \mathop {\lim }\limits_{h \to 0} \frac{{2k}}{{\sqrt {1 + k(0 - h)} + \sqrt {1 - k(0 - h)} }}$

$= \mathop {\lim }\limits_{h \to 0} \frac{{2k}}{{\sqrt {1 - kh} + \sqrt {1 + kh} }} = \frac{{2k}}{2} = k$

and $f(0) = \frac{{2 \times 0 + 1}}{{0 - 1}} = - 1$

$\Rightarrow$ $k = - 1$

Therefore the value of k = -1