MCQ
$\mathop {\lim }\limits_{x \to a} \frac{{\sqrt {3x - a} - \sqrt {x + a} }}{{x - a}} = $
- A$\sqrt 2 a$
- ✓$1/\sqrt {2a} $
- C$2a$
- D$1/2a$
$ = \mathop {\lim }\limits_{x \to a} \,\frac{{\sqrt {3x - a} - \sqrt {x + a} }}{{(x - a)}} \times \frac{{\sqrt {3x - a} + \sqrt {x + a} }}{{\sqrt {3x - a} + \sqrt {x + a} }}$
$ = \frac{2}{{2\sqrt {2a} }} = \frac{1}{{\sqrt {2a} }}$
Aliter : Apply $L$- Hospital’s rule
$\mathop {\lim }\limits_{x \to a} \,\frac{{\sqrt {3x - a} - \sqrt {x + a} }}{{x - a}} = \mathop {\lim }\limits_{x \to a} \,\frac{3}{{2\,\sqrt {3x - a} }} - \frac{1}{{2\,\sqrt {x + a} }}$
$ = \frac{3}{{2\sqrt {2a} }} - \frac{1}{{2\sqrt {2a} }} = \frac{1}{{\sqrt {2a} }}.$
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$\binom{s}{r}=\left\{\begin{array}{ll}\frac{s!}{r!(s-r)!} & \text { if } r \leq s \\ 0 & \text { if } r>s\end{array}\right.$
For positive integers $m$ and $n$, let
$(m, n) \sum_{ p =0}^{ m + n } \frac{ f ( m , n , p )}{\binom{ n + p }{ p }}$
where for any nonnegative integer $p$,
$f(m, n, p)=\sum_{i=0}^{ p }\binom{m}{i}\binom{n+i}{p}\binom{p+n}{p-i}$
Then which of the following statements is/are $TRUE$?
$(A)$ $(m, n)=g(n, m)$ for all positive integers $m, n$
$(B)$ $(m, n+1)=g(m+1, n)$ for all positive integers $m, n$
$(C)$ $(2 m, 2 n)=2 g(m, n)$ for all positive integers $m, n$
$(D)$ $(2 m, 2 n)=(g(m, n))^2$ for all positive integers $m, n$