If $\mathrm{f}(x)=3 x^{10}-7 x^8+5 x^6-21 x^3+3 x^2-7$, then $\lim _{\alpha \rightarrow 0} \frac{f(1-\alpha)-f(1)}{\alpha^3+3 \alpha}=$

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Aicharya · Accepted Answer

$\begin{aligned} & \mathrm{f}(x)=3 x^{10}-7 x^8+5 x^6-21 x^3+3 x^2-7 \ 	herefore \quad & \mathrm{f}^{\prime}(x)=30 x^9-56 x^7+30 x^5-63 x^2+6 x \ & \Rightarrow \mathrm{f}^{\prime}(1)=30-56+30-63+6=-53 \end{aligned}$ Now, $\lim _{\alpha ightarrow 0} \frac{f(1-\alpha)-f(1)}{\alpha^3+3 \alpha}$ $\begin{aligned} & =-\lim _{\alpha ightarrow 0} \frac{f(1-\alpha)-f(1)}{(1-\alpha)-1} 	imes \frac{1}{\alpha^2+3} \ & =-f^{\prime}(1) 	imes \frac{1}{3}=\frac{53}{3} \end{aligned}$

If $\mathrm{f}(x)=3 x^{10}-7 x^8+5 x^6-21 x^3+3 x^2-7$, then $\lim _{\alpha \rightarrow 0}…

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