Question

Evaluate n if Ltx-3 (Xn -3 n )/(X-3) =108​

Answer 1

Solution:

To Determine:- The value of n.

Given Equation :-

$\displaystyle \rm \longrightarrow \lim_{x \to3} \bigg( \dfrac{ {x}^{n} - {3}^{n} }{x - 3} \bigg) = 108$

We know that:-

$\displaystyle \rm \longrightarrow \lim_{x \to a} \dfrac{ {x}^{n} - {a}^{n} }{x - a} = n {a}^{n - 1}$

Therefore, the equation becomes :-

$\rm \longrightarrow n \times {3}^{n - 1} = 108$

$\rm \longrightarrow n \times {3}^{n - 1} =3 \times 3 \times 3 \times 4$

$\rm \longrightarrow n \times {3}^{n - 1} =4 \times {3}^{3}$

$\rm \longrightarrow n \times {3}^{n - 1} =4 \times {3}^{4 - 1}$

Comparing both sides, we get :-

$\rm \longrightarrow n = 4$

→ Therefore, the value of n satisfying the equation is 4.

Learn More:

Some standard limits.

$\displaystyle\rm 1.\:\: \lim_{x\to0}\sin(x)=0$

$\displaystyle\rm 2.\:\: \lim_{x\to0}\cos(x)=1$

$\displaystyle\rm 3.\:\: \lim_{x\to0}\dfrac{\sin(x)}{x}=1$

$\displaystyle\rm 4.\:\: \lim_{x\to0}\dfrac{\tan(x)}{x}=1$

$\displaystyle\rm 5.\:\: \lim_{x\to0}\dfrac{1-\cos(x)}{x}=0$

$\displaystyle\rm 6.\:\: \lim_{x\to0}\dfrac{\sin^{-1}(x)}{x}=1$

$\displaystyle\rm 7.\:\: \lim_{x\to0}\dfrac{\tan^{-1}(x)}{x}=1$

$\displaystyle\rm 8.\:\: \lim_{x\to0}\dfrac{\log(1+x)}{x}=1$

$\displaystyle\rm 9.\:\: \lim_{x\to0}\dfrac{e^{x}-1}{x}=1$