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Integration by Partial Fractions

The value of ...

Question

The value of $lim x \to 0 \frac{sin x^{n}}{(sin x)^{m}}, (m \leq n)$ , can be

A

0

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B

- 1

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C

1

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D

2

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Solution

The correct option is C $1$
Let, $L = lim x \to 0 \frac{sin x^{n}}{(sin x)^{m}}$
$= lim x \to 0 (\frac{sin x^{n}}{x^{n}}) (\frac{x^{n}}{x^{m}}) {(\frac{x}{sin x})}^{m}$
$= lim x \to 0 x^{n - m}$
So, if $m < n$
$L = 0,$ and
If $m = n$
$L = 1$

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Integration by Partial Fractions

Standard XII Mathematics

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