Question

If $\rm \displaystyle \lim_{x\to a}\ \frac{x^{3}-a^{3}}{x-a}=\displaystyle \lim_{x\to 1}\ \frac{x^{4}-1}{x-a}$, find all possible values of a.

Answer 1

Solution :

$\rm \displaystyle \lim_{x\to a}\ \frac{x^{3}-a^{3}}{x-a}=\displaystyle \lim_{x\to 1}\ \frac{x^{4}-a^{4}}{x-a}$

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We know that ,

$\rm \displaystyle \lim_{x\to a} \frac{x^{n}-a^{n}}{x-a}$

= $ n×a^{n-1}$

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=> $ 3a^{3-1} = 4a^{4-1} $

=> 3a² = 4a³

=> 3/4 = a³/a²

=> 3/4 = a

=> a = 3/4

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