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Question
Which of the following functions f has a removable discontinuity at x = x0? If the discontinuity is removable, find a function g that agrees with f for x ≠ x0 and is continuous on R.
`f(x) = (3 - sqrt(x))/(9 - x), x_0` = 9
Solution
The function f(x) is not defined at x = 9.
`lim_(x -> ) f(x) = lim_(x -> 9) (3 - sqrt(x))/(9 - x)`
= `lim_(x -> 9) (3 - sqrt(x))/(3^2 - (sqrt(x))^2`
=`lm_(x -> 9) (3 - sqrt(x))/((3+ sqrt(x))(3 - sqrt(x))`
= `lim_ (x -> 9) 1/(3 + sqrt(x))`
= `1/(3 + sqrt(9))`
= `1/(3 + 3)`
`lim_(x -> 9) f(x) = 1/6`
∴ Limit of the function f(x) exists at x = 9.
Hence, the function f(x) has a removable discontinuity at x = 9. Redefine the function f(x) as
`g(x) = {{:((3 - sqrt(x))/(9 - x), "if" x ≠ 9),(1/6, "if" x = 9):}`
Clearly, g(x) is defined at all points of R and is continuous on R.
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