Question

$\huge\red{\boxed{\sf QuEsTioN}}$


Find the positive integer “n” so that lim x → 3[(xn– 3n)/(x – 3)] = 108​

Answer 1

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

Given expression is

$\red{\rm :\longmapsto\:\displaystyle\lim_{x \to 3} \frac{ {x}^{n} - {3}^{n} }{x - 3} = 108}$

We know that,

$\boxed{ \rm \:\displaystyle\lim_{x \to a } \frac{ {x}^{n} - {a}^{n} }{x - a} = {na}^{n - 1}}$

So, using this, we get

$\rm :\longmapsto\: {n3}^{n - 1} = 108$

$\rm :\longmapsto\: {n3}^{n - 1} = 3 \times 3 \times 3 \times 4$

$\rm :\longmapsto\: {n3}^{n - 1} = 4 \times {3}^{3}$

$\rm :\longmapsto\: {n3}^{n - 1} = 4 \times {3}^{4 - 1}$

So, on comparing, we get

$\bf\implies \:n = 4$

Additional Information :-

$\boxed{ \rm \:\displaystyle\lim_{x \to 0} \frac{sinx}{x} = 1}$

$\boxed{ \rm \:\displaystyle\lim_{x \to 0} \frac{tanx}{x} = 1}$

$\boxed{ \rm \:\displaystyle\lim_{x \to 0} \frac{log(1 + x)}{x} = 1}$

$\boxed{ \rm \:\displaystyle\lim_{x \to 0} \frac{ {e}^{x} - 1}{x} = 1}$

$\boxed{ \rm \:\displaystyle\lim_{x \to 0} \frac{ {a}^{x} - 1}{x} = loga}$