For m-n - N- if x - 0limxn - sin xmxm - L- where L - n6- then m - n -

The correct option is B 0 lim_x→ 0x^n (sin x)^mx^m lim_x→ 0x^nx^m lim_x→ 0(sin xx)^m Since, limit is the finite quantity lim_x→ 0x^nx^m=1 ( ...

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Standard XII

Mathematics

Substitution Method to Remove Indeterminate Form

For m,n ∈ N...

Question

For $m, n \in N$ , if $lim x \to 0 \frac{x^{n} - (sin x)^{m}}{x^{m}} = L$ , where $L = \frac{n}{6}$ , then $m - n =$

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Solution

The correct option is B $0$
${lim}_{x \to 0} \frac{x^{n} - (sin x)^{m}}{x^{m}}$
${lim}_{x \to 0} \frac{x^{n}}{x^{m}} - {lim}_{x \to 0} (\frac{sin x}{x})^{m}$
Since, limit is the finite quantity
${lim}_{x \to 0} \frac{x^{n}}{x^{m}} = 1$ ( as ${lim}_{x \to 0} (\frac{sin x}{x})^{m} = 1$ )
$∴ n = m$
$\Rightarrow m - n = 0$

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For m,n∈N, if limx→0xn−(sinx)mxm=L, where L=n6, then m−n=

The correct option is B 0limx→ 0xn−(sinx)mxmlimx→ 0xnxm−limx→ 0(sinxx)mSince, limit is the finite quantitylimx→ 0xnxm=1( as limx→ 0(sinxx)m=1)∴n=m⇒m−n=0

For $m, n \in N$ , if $lim x \to 0 \frac{x^{n} - (sin x)^{m}}{x^{m}} = L$ , where $L = \frac{n}{6}$ , then $m - n =$