Question

lim x tends to infinity
$\frac{(3x + 3)!}{(x + 1)^{3}(3x)!}$
​

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:\displaystyle \lim_{x \: \to \: \infty } \: \sf \: \dfrac{(3x + 3)!}{(x + 1)^{3}(3x)!}$

$\rm \: \: = \: \displaystyle \lim_{x \: \to \: \infty } \: \sf \: \dfrac{(3x + 3)(3x + 2)(3x + 1) \: \cancel{(3x)!}}{(x + 1)^{3} \cancel{(3x)!}}$

$\rm \: \: = \: \displaystyle \lim_{x \: \to \: \infty } \: \sf \: \dfrac{(3x + 3)(3x + 2)(3x + 1)}{(x + 1)^{3}}$

$\rm \: \: = \: \displaystyle \lim_{x \: \to \: \infty } \: \sf \: \dfrac{x \bigg(3 + \dfrac{3}{x} \bigg)x \bigg(3 + \dfrac{2}{x} \bigg)x \bigg(3 + \dfrac{1}{x} \bigg)}{ {x}^{3} \bigg(1 + \dfrac{1}{x} \bigg)^{3}}$

$\rm \: \: = \: \displaystyle \lim_{x \: \to \: \infty } \: \sf \: \dfrac{ \bigg(3 + \dfrac{3}{x} \bigg) \bigg(3 + \dfrac{2}{x} \bigg) \bigg(3 + \dfrac{1}{x} \bigg)}{ \bigg(1 + \dfrac{1}{x} \bigg)^{3}}$

$\rm \: \: = \: \dfrac{(3 + 0)\times (3 + 0) \times (3 + 0)}{ {(1 + 0)}^{3} }$

$\rm \: \: = \: \dfrac{3 \times 3 \times 3}{1}$

$\rm \: \: = \: 27$

Hence,

$\: \: \: \: \: \: \: \red{ \boxed{\bf :\longmapsto\:\displaystyle \lim_{x \: \to \: \infty } \: \bf \: \dfrac{(3x + 3)!}{(x + 1)^{3}(3x)!} = 27}}$

Additional Information :-

$\begin{gathered}(1)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{sin \: x}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(2)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{tan \: x}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(3)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{tan^{ - 1} \: x}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(4)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{sin^{ - 1} \: x}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(5)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{log(1 \: + \: x)}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(6)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{ {e}^{x} - 1 }{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(7)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: { \bigg(x + 1 \bigg)}^{ \dfrac{1}{x} } \: = \: e }}}}}} \\ \end{gathered}$

$\begin{gathered}(8)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{ {a}^{x} - 1 }{x} \: = \: log(a) }}}}}} \\ \end{gathered}$