If limx→0pcosx +xe1x1+sin x+q cos x.e1x=0, then which of the following is/are incorrect about p, q ?

limx→0+pcosxe1x +x1+sin xe1x+q cos x=0 ⇒q∈R−{0} limx→0-pcosx+xe+1x1+sin x+q cos xe1x⇒p=0

If limx→0pcosx +xe1x1+sin x+q cos x.e1x=0, then which of the following is/are incorrect about p, q ?

limx→0+pcosxe1x +x1+sin xe1x+q cos x=0 ⇒q∈R−{0} limx→0-pcosx+xe+1x1+sin x+q cos xe1x⇒p=0

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If $\lim_{x \to 0} \frac{p \cos x + x e^{\frac{1}{x}}}{1 + \sin x + q \cos x . e^{\frac{1}{x}}} = 0$ , then which of the following is/are incorrect about p, q ?

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$p = 0, q \in R$

$p = 4, q = 2$

$p = 2, q \in R$

$p = 0, q = 2$

answer is .

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Detailed Solution

$\lim_{x \to 0^{+}} \frac{\frac{p \cos x}{e^{\frac{1}{x}}} + x}{\frac{1 + \sin x}{e^{\frac{1}{x}}} + q \cos x} = 0 \Rightarrow q \in R - {0}$ $\lim_{x \to 0^{-}} \frac{p \cos x + x e^{\frac{+ 1}{x}}}{1 + \sin x + q \cos x e^{\frac{1}{x}}} \Rightarrow p = 0$