Where have I gone wrong in evaluating $\lim_{x \to \infty}\sqrt x(\sqrt{x+c}- \sqrt x )$? 
Evaluate 
  $$\lim_{x \to \infty}\sqrt x(\sqrt{x+c}- \sqrt x )$$

Attempt:
$$\begin{align} 
\lim_{x \to \infty}\sqrt x(\sqrt{x+c}- \sqrt x ) &= \lim_{x\to \infty }(\sqrt{x^2+cx}- x) \\
&= \lim _{x\to \infty}x\left(\sqrt{\left(1+\dfrac{c}{x}\right)}-1\right)\\
&= \lim _{x \to \infty} x \times 0 \\
&= 0 \times \infty \\
&=0
\end{align}$$
But the answer given is :

 $$\frac c 2$$


Write like this (using $a^2-b^2=(a-b)(a+b)$)
$$ \sqrt{x^2+cx} - x = \frac{ x^2+cx - x^2}{\sqrt{x^2+cx}+x}  = \frac{cx}{\sqrt{x^2+cx}+x}$$
Now, factor $x$ numerator and denominator cancel and we obtain 
$$ \frac{ c }{ \sqrt{ 1 + \frac{c}{x} } + 1 } $$
