Question

Let $f\left(x\right)=3x^{10}-7x^8+5x^6-21x^3+3x^2-7$, then the value of $\underset{h\to 0}{lim}\frac{f(1-h)-f\left(1\right)}{h^3+3h}$ is:

• A. $\frac{50}{3}$
• B. $\frac{22}{3}$
• C. $13$
• D. $\frac{53}{3}$

Answer 1

Answer: (D)

$\frac{53}{3}$

$L=\underset{h\to 0}{lim}\frac{f(1-h)-f\left(1\right)}{h^3+3h}$
$L=\underset{h\to 0}{lim}\frac{f(1-h)-f\left(1\right)}{h(h^2+3)}$
$=\underset{h\to 0}{lim}\frac{f(1-h)-f\left(1\right)}{-h}\underset{h\to 0}{lim}\frac{-1}{(h^2+3)}$
$L=f^{\prime }\left(1\right)(-\frac{1}{3})$ .....(1)
Since, $f\left(x\right)=3x^{10}-7x^8+5x^6-21x^3+3x^2-7$
$f^{\prime }\left(x\right)=30x^9-56x^7+30x^5-63x^2+6x$
$\Rightarrow f^{\prime }\left(1\right)=-53$
Put this value in (1), we get
$\Rightarrow L=\frac{53}{3}$

Hence, option D.