Let `f(x)=3x^10-7x^8+5x^6-21x^3+3x^2-7` , then the value of `lim_(h->0) (f(1-h)-f(1))/(h^3+3h)`

Options

A. `(-55)/(3)`
B. `(53)/(3)`
C. `-(53)/(3)`
D. `(55)/(3)`

Answer

B

Explanation

`lim_(xto0)(f(1-alpha)-f(1))/(alpha^3+3alpha)`
`lim_(xto0)(f(1-alpha)-f(1))/({(1-alpha)-1}(alpha^2+3))`
`lim_(xto0)(f(1-alpha)-f(1))/({(1-alpha)-1})xx(1)/(alpha^2+3)`
`=-f'(1)xx(1)/(3)=-(1)/(3)f'(1)`
Now, `f(x)=3x^10-7x^8+5x^6-21x^3+3x^2-7`
`rArr f'(x)=30x^9-56x^7+30x^5-63x^2+6x`
`rArr f'(1)=30-56+30-63+6=-53`
Hence, `lim_(xto 0) (f'(1-alpha)-f(1))/(alpha^3+3alpha)=(53)/(3)`