If `g(x)=(lim)_(mvecoo)(x^mf(x)+h(x)+3)/(2x^m+4x+1)` when `x!=1a n dg(1)=e^3` such that `f(x),g(x)a n dh(x)` are continuous functions at `x=1` then the value of `5f(1)-2h(1)` is `7` b. `6` c. `9` d. `8`

Options

A. 7
B. 6
C. 9
D. 8

Answer

B

Explanation

`underset(xrarr1^(+))(lim)g(x)=underset(xrarr1^(+))(lim){underset(mrarroo)(lim)(x^(m)f(x)+h(x)+3)/(2x^(m)+4x+1)}`
`=underset(mrarroo)(lim)underset(xrarr1^(+))(lim){(x^(m)f(x)+h(x)+3)/(2x^(m)+4x+1)}`
`=underset(mrarroo)(lim)underset(xrarr1^(+))(lim){(f(x)+(h(x)+3)/(x^(m)))/(2+(4x+1)/(x^(m)))}`
`=(f(1))/(2)`
`underset(xrarr1^(-))(lim)g(x)=underset(xrarr1^(-))(lim){underset(mrarroo)(lim)(x^(m)f(x)+g(x)+3)/(2x^(m)+4x+1)}`
`=underset(mrarroo)(lim)underset(xrarr1^(-))(lim){(x^(m)f(x)+h(x)+3)/(2x^(m)+4x+1)}`
`=(h(1)+3)/(5)`
g(x) is continuous at x = 1
`therefore" "(f(1))/(2)=(h(1)+3)/(5)`
`rArr" "5f(1)-2h(1)=6`