MCQ
$\mathop {{\rm{lim}}}\limits_{x \to 0} \frac{{\left( {1 - cos2x} \right)\left( {3 + \cos x} \right)}}{{x\;tan4x}}$ =
- A$ - \frac{1}{4}$
- B$\frac{1}{2}$
- C$1$
- ✓$2$
$ = \mathop {\lim }\limits_{x \to 0} \frac{{2{{\sin }^2}x\left( {3 + \cos x} \right)}}{{x \times \frac{{\tan 4x}}{{4x}} \times 4x}}$
$ = \mathop {\lim }\limits_{x \to 0} \frac{{2{{\sin }^2}x}}{{{x^2}}} \times \mathop {\lim }\limits_{x \to 0} \frac{{\left( {3 + \cos x} \right)}}{4} \times \frac{1}{{\mathop {\lim }\limits_{x \to 0} \frac{{\tan 4x}}{{4x}}}}$
$ = 2 \times \frac{4}{4} \times 1$ ($\because $ $\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{\theta } = 1$ and $\mathop {\lim }\limits_{\theta \to 0} \frac{{\tan \theta }}{\theta } = 1$)
$=2$
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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 $m, n$