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The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous : f(x) = (3-8x3-2x)1x, for x ≠ 0

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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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Chapter 8: Continuity - EXERCISE 8.1 [Page 174]

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