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

Answer 1

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.