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Question

f(x)-Ikx +1,ไม-5if5if x > 529,at 5at X-,

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Solution

The expression for the function f is defined as,

f( x )={ kx+1,ifx5 3x5,ifx>5 }

The left hand limit of the function at x=5 is,

lim x 5 f( x )= lim x 5 ( kx+1 ) = lim x5h ( kx+1 ) = lim h0 ( k( 5h )+1 ) =5k+1

The right hand limit of the function at x=5 is,

lim x 5 + f( x )= lim x 5 + ( 3x5 ) = lim x5+h ( 3x5 ) = lim h0 ( 3( 5+h )5 ) = lim h0 ( 15+3h5 )

Solve for the right hand limit.

lim x 5 + f( x )=155 =10

The exact value of the function for x=5is,

f( x=5 )=kx+1 =5k+1

The condition for continuity of the function f at x=5 is fulfilled if left hand limit, right hand limit and the value at that specified point are equal.

It is observed that,

lim x 5 f( x )= lim x 5 + f( x )=f( x=5 ) 5k+1=10=5k+1 5k=9 k= 9 5


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