If lim x - 3 x n -3 n - x -3-108- find the value of n-

Lim_x → 3x^n 3^n/x 3 =108 n(3)^n 1=108 = [ ∵lim_x → ax^n a^n/x a=na^n 1 ] n(3)^n 1=2 × 2 × 3 × 3 × 3 =4(3)^3 n(3)^n 1=4(3)^4 1 Comparing we get n =4 ...

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Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

If lim x → 3 ...

Question

If ${lim}_{x \to 3} \frac{x^{n} - 3^{n}}{x - 3} = 108$ , find the value of n.

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Solution

${lim}_{x \to 3} \frac{x^{n} - 3^{n}}{x - 3} = 108$

$n (3)^{n - 1} = 108$

$= [∵ {lim}_{x \to a} \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}]$

$n (3)^{n - 1} = 2 \times 2 \times 3 \times 3 \times 3$

$= 4 (3)^{3}$

$n (3)^{n - 1} = 4 (3)^{4 - 1}$

Comparing we get

n =4

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If limx→3xn−3nx−3=108, find the value of n.

limx→3xn−3nx−3=108 n(3)n−1=108 =[∵limx→axn−anx−a=nan−1] n(3)n−1=2×2×3×3×3 =4(3)3 n(3)n−1=4(3)4−1 Comparing we get n =4

If ${lim}_{x \to 3} \frac{x^{n} - 3^{n}}{x - 3} = 108$ , find the value of n.

${lim}_{x \to 3} \frac{x^{n} - 3^{n}}{x - 3} = 108$

$n (3)^{n - 1} = 108$

$= [∵ {lim}_{x \to a} \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}]$

$n (3)^{n - 1} = 2 \times 2 \times 3 \times 3 \times 3$

$= 4 (3)^{3}$

$n (3)^{n - 1} = 4 (3)^{4 - 1}$

Comparing we get

n =4