Question

Lim [xn-4n/x-4]=48 and n€N, find n.
X-->4

Answer 1

$\begin{gathered}\Large{\bold{{\underline{Formula \: Used - }}}}  \end{gathered}$

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

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

Given that,

$\rm :\longmapsto\: \displaystyle\bf \:\lim_{x\to 4} \: \dfrac{ {x}^{n} - {4}^{n}}{x - 4} = 48$

Using the above result, we get

$\rm :\longmapsto\: {n(4)}^{n - 1} = 48$

$\rm :\longmapsto\: {n(4)}^{n - 1} = 4 \times 4 \times 3$

$\rm :\longmapsto\: {n(4)}^{n - 1} = {4}^{2} \times 3$

$\rm :\longmapsto\: {n(4)}^{n - 1} = {(4)}^{3 - 1} \times 3$

So, on comparing we get

$\rm :\implies\:n \: = \: 3$

Additional Information :-

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

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

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

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

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

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

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