Question

Suppose f(x) ={ a+bx, x\(<\)1 4, x=1 b-ax x\(>\)1 and if lim x \(\rightarrow\)1 f(x) = f(1), what are possible values of a and b?

Answer 1

Given:

f(x) = a + bx,   x < 1
f(1) = 4
f(x) = b − ax,   x > 1

Given that,
lim_x→1 f(x) = f(1)

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Step 1: Evaluate Left Hand Limit (LHL)

LHL = lim_x→1⁻ f(x)

= lim_x→1⁻ (a + bx)

= a + b

Since limit exists and equals f(1),

a + b = 4   …(1)

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Step 2: Evaluate Right Hand Limit (RHL)

RHL = lim_x→1⁺ f(x)

= lim_x→1⁺ (b − ax)

= b − a

Since limit exists and equals f(1),

b − a = 4   …(2)

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Step 3: Solve equations (1) and (2)

a + b = 4
b − a = 4

Adding both equations:

2b = 8

b = 4

Substitute b = 4 in equation (1):

a + 4 = 4

a = 0

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Final Answer:

The possible values are:
a = 0 and b = 4