Question

x< 032. )xm Osrl. For what integers m and n does both lim f(x)x→0nx +m,and lim f (x) exist?x→1

Answer 1

Let the given function defined over their range as

f( x )={ m x 2 +n,x<0 nx+m,0≤x≤1 n x 3 +m,x>1

We need to take a common point at x=0 and x=1 and find the left hand and right hand limit of the function.

From the definition of limits, we know that:

lim x→a f( x )=f( a )

For x=0 , left and right hand limits are:

lim x→ 0 − f( x )= lim x→0 ( m x 2 +n ) =m ( 0 ) 2 +n =n (1)

lim x→ 0 + f( x )= lim x→0 ( nx+m ) =0⋅n+m =m (2)

From equations (1) and (2) we can conclude that lim x→0 f( x ) exists, if m=n .

For x=1 , left and right hand limits are

lim x→ 1 − f( x )= lim x→1 ( nx+m ) =( n⋅1+m ) =( m+n ) (3)

lim x→ 1 + f( x )= lim x→1 ( n x 3 +m ) =n⋅ 1 3 +m =( m+n ) (4)

Since, lim x→ 1 − f( x )= lim x→ 1 + f( x )

Thus, from equations (3) and (4) we can conclude that lim x→1 f( x ) exists for the integral values of m and n .