Question

limit x tends to a x^m-a^m/x^n-a^n​

Answer 1

SOLUTION

TO EVALUATE

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

FORMULA TO BE IMPLEMENTED

We are aware of the formula that

$\displaystyle \lim_{x \to a} \: \frac{{x}^{m} - {a}^{m} }{x - a} = m \: \times {a}^{m - 1}$

EVALUATION

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

$\displaystyle = \: \frac{ \displaystyle \lim_{x \to a} \: \frac{{x}^{m} - {a}^{m} }{x - a} }{ \displaystyle \lim_{x \to a} \: \frac{{x}^{n} - {a}^{n} }{x - a}}$

$\displaystyle = \: \frac{m \: \times {a}^{m - 1} }{n \times {a}^{n - 1} }$

$\displaystyle = \: \frac{m \: \times {a}^{(m - 1 - n + 1)} }{n }$

$\displaystyle = \: \frac{m \: \times {a}^{(m - n )} }{n }$

$\displaystyle = \: \frac{m } {n } \times {a}^{(m - n )}$

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