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Question

Suppose f(x)=a+bx,x<14,x=1bax,x>1 and if limx1f(x)=f(1) what are possible values of a and b ?

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Solution

Step 1: Limit at x=1
Here, limit exists at x1
i.e., L.H.L.=R.H.L.=f(1)=4 (i)
L.H.L.=limx1f(x)=limh0f(1h)
L.H.L.=limh0a+b(1h)
L.H.L.=a+b(10)
L.H.L.=a+b (ii)


R.H.L.=limx1+f(x)=limh0f(1+h)
R.H.L.=limh0ba(1+h)
R.H.L.=ba(1+0)
R.H.L.=ba (iii)

Step 2: Solve equations for value of a and b
Given, limx1f(x)=f(1)
limx1f(x)=limx1+f(x)=f(1)
a+b=ba=4

From (ii) and (iii)
a+b=4 (iv)
ba=4 (v)
Adding equation (iv) and (v)
a+b+ba=4+4
2b=8b=4
Also, a+b=4
a+4=4
a=0


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