Question

If $f(x) = 3x^{10} - 7x^{8} + 5x^{6} - 21x^{3} + 3x^{2} - 7,$ then $\displaystyle \lim_{\alpha \to 0}$$\frac{f\left(1-\alpha\right)-f \left(1\right)}{\alpha^{3}+3\alpha}$ is

• A. $-\frac{53}{3}$
• B. $\frac{53}{3}$
• C. $-\frac{55}{3}$
• D. $\frac{55}{3}$

Answer 1

Answer: (B)

Let $f'(x) = 3x^{10} - 7x^{8} + 5x^{6} - 21x^{3} + 3x^{2} - 7,$
$f '(x)=30x^9-56x^7+30x^5-63x^2+6x$
$f ' (1)=30-56+30-63+ 6$
$= 66-63-56 = -53$
Consider $\displaystyle \lim_{\alpha \to 0}$$\frac{f \left(1-\alpha\right)-f \left(1\right)}{\alpha^{3}+3\alpha}$
$=\displaystyle \lim_{\alpha \to 0}$ $\frac{f '\left(1-\alpha\right)-\left(1\right)-0}{3\alpha^{3}+3}$ (By using L’hospital rule)
$\frac{f '\left(1-0\right)-\left(1\right)}{3\left(0\right)^{2}+3}=-\frac{f '\left(1\right)}{3}=\frac{53}{3}$