MCQ
Let $p(x)$ be a polynomial such that $p(x)-p^{\prime}(x)=x^n$, where $n$ is a positive integer. Then, $p(0)$ equals
- ✓$n !$
- B$(n-1) !$
- C$\frac{1}{n !}$
- D$\frac{1}{(n-1) !}$
Let
$p =\lim _{x \rightarrow 0}\left(\frac{x}{\sin x}\right)^{6 / x^2}$
$\Rightarrow \quad \log p =\lim _{x \rightarrow 0} \frac{6}{x^2} \log \left(\frac{x}{\sin x}\right)$
$\Rightarrow \quad \log p =\lim _{x \rightarrow 0} \frac{6 \log \left(\frac{x}{\sin x}\right)}{x^2}$
Apply $L-$ Hospital rule
$\log p=\lim _{x \rightarrow 0}$
$6 \frac{\frac{\sin x}{x} \frac{(\sin x-x \cos x)}{\sin ^2 x}}{2 x}$
$\log p=\lim _{x \rightarrow 0} 6 \frac{\sin x(\sin x-x \cos x)}{x \cdot 2 x \sin ^2 x}$
$\log p=\lim _{x \rightarrow 0} 3 \frac{\sin x}{x} \times \lim _{x \rightarrow 0}$
$\frac{\sin x-x \cos x}{\frac{\sin ^2 x}{x^2}} \times x^3$
$\log p=3 \times \lim _{x \rightarrow 0} \frac{\sin x-x \cos x}{x^3}$
$\Rightarrow \quad \log p=3 \times \lim _{x \rightarrow 0} \frac{\sin x-x \cos x}{x^3}$
Again apply L.Hospital's rule
$\log p=3 \lim _{x \rightarrow 0} \frac{\cos x-\cos x+x \sin x}{3 x^2}$
$\Rightarrow \log p=\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
$\therefore \quad p=e^1=e$
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has infinitely many solutions, is: