First slide

Introduction to limits

Question

The value of $lim_{x \to a} \frac{x^{m} - a^{m}}{x^{n} - a^{n}}$ , is

Moderate

A

$\frac{m}{n} a^{m - n}$
B

$\frac{n}{m} a^{m - n}$
C

$\frac{m}{n} a^{n + n}$
D

$\frac{m}{n}$

Solution

We have,
$lim_{x \to a} \frac{x^{n} - a^{n}}{x^{n} - a^{n}}$ [from $\frac{0}{0}$ ]
$\begin{array}{l} = lim_{x \to a} \{\frac{x^{m} - a^{m}}{x - a} \times \frac{x - a}{x^{n} - a^{n}}\} \\ = lim_{x \to a} \{\frac{x^{n} - a^{m}}{x - a} \div \frac{x^{n} - a^{n}}{x - a}\} \\ = lim_{x \to a} \frac{x^{n} - a^{m}}{x - a} \div lim_{x \to a} \frac{x^{n} - a^{n}}{x - a} = m a^{m - 1} \div n a^{n - 1} = \frac{m}{n} a^{m - n} \end{array}$

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