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Question
The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :
f(x) = `((3 - 8x)/(3 - 2x))^(1/x)`, for x ≠ 0
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Solution
f(x) = `((3 - 8x)/(3 - 2x))^(1/x)`, for x ≠ 0
Here, f(0) is not defined.
Consider, `lim_(x -> 0) "f"(x) = lim_(x -> 0) ((3 - 8x)/(3 - 2x))^(1/x)`
= `lim_(x -> 0) [(3(1 - (8x)/3))/(3(1 - (2x)/3))]^(1/x)`
= `lim_(x -> 0) ((1 - (8x)/3)^(1/x))/(1 - (2x)/3)^(1/x)`
= `(lim_(x -> 0) [(1 - (8x)/3)^((-3)/(8x))]^((-8)/3))/(lim_(x -> 0) [(1 - (2x)/3)^((-3)/(2x))]^((-2)/3))`
= `("e"^((-8)/3))/("e"^((-2)/3)) ...[(because x -> 0"," (-8x)/3 -> 0"," (-2x)/3 -> 0),(and lim_(x -> 0) (1 + x)^(1/x) = "e")]`
= `"e"^((-6)/3)`
= e–2
∴ `lim_(x -> 0) "f"(x)` exists
But f(0) is not defined.
∴ f(x) has a removable discontinuity at x = 0.
This discontinuity can be removed by defining f(0) = e–2
∴ f(x) can be redefined as
f(x) `{:(= ((3 - 8x)/(3 - 2x))^(1/x), ";" x ≠ 0),(= "e"^(-2), ";" x = 0):}`
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