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Q.
Let $f(x)=3 x^{10}-7 x^{8}+5 x^{6}-21 x^{3}+3 x^{2}-7$. Then find the value of $\displaystyle\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{h^{3}+3 h}$.
Limits and Derivatives
Solution:
L Hospital rule:
given limit is $f'(1)\left(-\frac{1}{3}\right)=\frac{53}{3}$
$f'(x)=30 x^{9}-56 x^{7}+30 x^{5}-63 x^{2}+6 x$
$f'(1)=-53$
So, $-\frac{ f '(1)}{3}=\frac{53}{3}$