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Suppose f x equals a plus bx , for x less than 1 and if 1 limit x tends to 1 f (x) = f (1) what are possible values of a and b?

28.   Suppose 

f (x) = \left\{\begin{matrix} a+bx & x < 1 \\ 4 & x = 1 \\ b - ax & x > 1 \end{matrix}\right.   f (x) = f (1) what are possible values of a and b?

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Given,

f (x) = \left\{\begin{matrix} a+bx & x < 1 \\ 4 & x = 1 \\ b - ax & x > 1 \end{matrix}\right.

And

 \lim_{x\rightarrow 1} f(x)=f(1)

Since the limit exists,

left-hand limit = Right-hand limit = f(1).

Left-hand limit  = f(1)

\lim_{x\rightarrow 1^-} f(x)= \lim_{x\rightarrow 1}(a+bx)=a+b(1)=a+b=4

Right-hand limit

\lim_{x\rightarrow 1^+} f(x)= \lim_{x\rightarrow 1}(b-ax)=b-a(1)=b-a=4

From both equations, we get that,

a=0 and b=4

Hence the possible value of a and b are 0 and 4 respectively.

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