Choose the correct answer. _ x 0 x^ m -1 x^ n -1 is equal to: - Vidyadip

MCQ

Choose the correct answer. $\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}^{\text{m}}-1}{\text{x}^{\text{n}}-1}$ is equal to:

A
$1$
✓
$\frac{\text{m}}{\text{n}}$
C
$\frac{-\text{m}}{\text{n}}$
D
$\text{m}^{2}\text{n}^{2}$

✓

Answer

Correct option: B.

$\frac{\text{m}}{\text{n}}$

Given $\lim\limits_{\text{x} \rightarrow 1}\frac{\text{x}^{\text{m}}-1}{\text{x}^{\text{n}}-1}=\lim\limits_{\text{x} \rightarrow 1}\frac{\frac{\text{x}^{\text{m}-(1)^\text{m}}}{\text{x}-1}}{\frac{\text{x}^{\text{n}-(1)^{\text{n}}}}{\text{x}-1}}$
$=\frac{\text{m}(1)^{\text{m}-1}}{\text{n}(1)^{\text{n}-1}}=\frac{\text{m}}{\text{n}}$
$=\frac{\text{m}}{\text{n}}$

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