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A function f: RvecR is defined as f(x)=(...

A function `f: RvecR` is defined as `f(x)=(lim)_(nvecoo)(a x^2+b x+c+e^(n x))/(1+cdote^(n x))` is continuous on then Point lies on the space Point represents the 2-dimensional Cartesian plane Locus of point `(a ,c)a n d(c ,b)` intersect at one point Point `(a ,b ,c)` lies on the plane in space

A

point (a, b, c) lies on line in space

B

point (a, b) represents the 2-dimensional Cartesian plane

C

Locus of point (a, c) and (c, b) intersect at one point

D

point (a, b, c) lies on the plane in space

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`f(x)=underset(nrarroo)(lim)(ax^(2)+bx+c+e^(nx))/(1+c.e^(nx))`
`={{:(underset(nrarroo)(lim)(ax^(2)+bx+c+(e^(x))^(n))/(1+c.(e^(x))^(n)),,xlt0),(underset(nrarroo)(lim)(ax^(2)+bx+c+(e^(x))^(n))/(1+c.(e^(x))^(n)),,x=0),(underset(nrarroo)(lim)((ax^(2)+bx+c)/((e^(x))^(n)))/((1)/((e^(x))^(n))+c),,xgt0):}`
`={{:(ax^(2)+bx+c,,xlt0),(1,,x=0),((1)/(c),,xgt0):}`
since f(x) is continuous function `AA x in R`
`therefore" "underset(xrarr0^(+))(lim)f(x)=underset(xrarr0^(-))(lim)f(x)=f(0)`
`rArr" "underset(xrarr0^(+))(lim)((1)/(c))=underset(xrarr0^(-))(lim)(ax^(2)+bx+c)=1`
`rArr" "(1)/(c)=1=c rArr c=1`
`therefore " "c=1, a, b in R`
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