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Question
Suppose f(x) = `{(a+bx, x < 1),(4, x = 1),(b-ax, x > 1):}` and if `lim_(x -> 1) f(x) = f(1)` what are possible values of a and b?
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Solution
When x < 1, f(x) = a + bx
To find the limit of the left side, keeping the value of x less than 1 and near 1 in f(x),
| x | 0.99 | 0.999 | 0.9999 |
| f(x) | a + 1.99b | a + 0.999b | a + 0.9999b |
∴ `lim_(x → 1^-) f(x) = a + b`
To find the limit of the right-hand side, f(x) = b − ax, with values of x greater than 1 and near 1.
| x | 1.01 | 1.0001 | 1.00001 |
| f(x) | b − 1.01a | b − 1.0001a | b − 1.00001a |
∴ `lim_(x → 1^+) f(x) = b - a`
∴ If limit exists at x = 1 then
`lim_(x → 1^-) f(x) = a + b = b + a = f(1) = 4`
∴ b + a = 4 .....(i)
`lim_(x → 1^+) f(x) = b - a = f(1) = 4`
∴ b − a = 4 ....(ii)
On adding equations (i) and (ii),
2b = 8 or b = 4
Putting b = 4 in equation (i),
4 + a = 4 or a = 0
Hence, a = 0, b = 4
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