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Let f(x)=3x^10-7x^8+5x^6-21x^3+3x^2-7 ,...

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)`

A

`(53)/(3)`

B

`(22)/(3)`

C

`13`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
` lim_(xto0) (f(1-h)-f(1))/(h^3+3h)=lim_(xto0) (f(1-h)-f(1))/(-h)xx(-1)/(h^2+3)`
` =lim_(hto0) (f(1-h)-f(1))/((1-h)-1) xx (-1)/(h^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_(xto0)(f(1-h)-f(1))/(h^3+3h)=-(1)/(3)xx-53=(53)/(3)`.
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