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Question: The value of \(\lim_{x \rightarrow a}\frac{x^{n} - a^{n}}{x - a}\) is...

The value of lim⁡x→axn−anx−a\lim_{x \rightarrow a}\frac{x^{n} - a^{n}}{x - a} is

A

0

B

nan−1na^{n - 1}

C

nanna^{n}

D

1

Answer

nan−1na^{n - 1}

Explanation

Solution

lim⁡x→axn−anx−a\lim_{x \rightarrow a}\frac{x^{n} - a^{n}}{x - a} = lim⁡x→a(x−a)(xn−1+xn−2a+..+an−1)(x−a)\lim_{x \rightarrow a}\frac{(x - a)(x^{n - 1} + x^{n - 2}a + .. + a^{n - 1})}{(x - a)}

= lim⁡x→a(xn−1+xn−2a+....+an−1)\lim_{x \rightarrow a}(x^{n - 1} + x^{n - 2}a + .... + a^{n - 1}) = n.an−1n.a^{n - 1}.