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If `f(x)` is a differentiable function such that `f\'(x)=7-(3)/(4)(f(x))/(x)`, `f(1)!=4`, then `lim_(xrarr0^+)xf((1)/(x))` is equal to (a) does not exist (b) exist and equal to 4 (c) exist and is equal to `(4)/(7)` (d) exists and equal to 0

A

exists and equals `4//7`

B

exists and equals 0

C

exist and equals 4

D

does not exist

Text Solution

Verified by Experts

The correct Answer is:
C
Let `f(x)=y`
`(dy)/(dx)=7-(3)/(4)(y)/(x)(x gt 0)`
`(dy)/(dx)+(3)/(4)(y)/(x)=7`
If `e^(int(3)/(4x))=x^(3//4)`
`yx^(3//4)=7 int x^(3//4) dx+c`
`yx^(3//4)=4x^(7//4)+c`
`y=4x+cx^(-3//4)`
`f((1)/(x))=(4)/(x)+cx^(3//4)` `xf((1)/(x))=4+cx^(7//4)`
`lim_(x to 0^(+)) xf ((1)/(x))=4`
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