Let $f ( x )$ be a continuously differentiable function on the interval $( 0 , \infty )$ such that $f ( 1 ) = 2$ and $$\lim _ { t \rightarrow x } \frac { t ^ { 10 } f ( x ) - x ^ { 10 } f ( t ) } { t ^ { 9 } - x ^ { 9 } } = 1$$ for each $x > 0$. Then, for all $x > 0 , f ( x )$ is equal to (A) $\frac { 31 } { 11 x } - \frac { 9 } { 11 } x ^ { 10 }$ (B) $\frac { 9 } { 11 x } + \frac { 13 } { 11 } x ^ { 10 }$ (C) $\frac { - 9 } { 11 x } + \frac { 31 } { 11 } x ^ { 10 }$ (D) $\frac { 13 } { 11 x } + \frac { 9 } { 11 } x ^ { 10 }$
Let $f ( x )$ be a continuously differentiable function on the interval $( 0 , \infty )$ such that $f ( 1 ) = 2$ and
$$\lim _ { t \rightarrow x } \frac { t ^ { 10 } f ( x ) - x ^ { 10 } f ( t ) } { t ^ { 9 } - x ^ { 9 } } = 1$$
for each $x > 0$. Then, for all $x > 0 , f ( x )$ is equal to
(A) $\frac { 31 } { 11 x } - \frac { 9 } { 11 } x ^ { 10 }$
(B) $\frac { 9 } { 11 x } + \frac { 13 } { 11 } x ^ { 10 }$
(C) $\frac { - 9 } { 11 x } + \frac { 31 } { 11 } x ^ { 10 }$
(D) $\frac { 13 } { 11 x } + \frac { 9 } { 11 } x ^ { 10 }$