Question

If f(x) = 3x¹⁰ – 7x⁸ + 5x⁶ – 21x³ + 3x² – 7, then `lim_(α rightarrow 0) (f(1 - α) - f(1))/(α^3 + 3α)` is ______. 

पर्याय

• `-53/3`

• `53/3`

• `-55/3`

• `55/3`

Answer 1

If f(x) = 3x¹⁰ – 7x⁸ + 5x⁶ – 21x³ + 3x² – 7, then `lim_(α rightarrow 0) (f(1 - α) - f(1))/(α^3 + 3α)` is `underlinebb(53/3)`. 

Explanation:

Let f(x) = 3x¹⁰ – 7x⁸ + 5x⁶ – 21x³ + 3x² – 7

f'(x) = 30x⁹ – 56x⁷ + 30x⁵ – 63x² + 6x

f'(1) = 30 – 56 + 30 – 63 + 6

= 66 – 63 – 56

= –53

Consider `lim_(α rightarrow 0) (f(1 - α) - f(1))/(α^3 + 3α)`

= `lim_(α rightarrow 0) (f^'(1 - α)(-1) - 0)/(3α^2 + 3)`  ...(By using L’Hospital rule)

= `(f^'(1 - 0)(-1))/(3(0)^2 + 3)`

= `(-f^'(1))/3`

= `53/3`