Question

f(x) = `(sqrt(x + 3) - 2)/(x^3 - 1)`  for x ≠ 1

= 2    for x = 1, at x = 1.

Answer 1

f(1) = 2  …[given]

`lim_(x→1) "f"(x) = lim_(x→1) (sqrt(x + 3) - 2)/(x^3 - 1)`

= `lim_(x→1) ((sqrt(x + 3) - 2)/(x^3 - 1) xx (sqrt(x + 3) + 2)/(sqrt(x + 3) + 2))`

= `lim_(x→1) ((x + 3 - 4)/((x^3 - 1)(sqrt(x + 3) + 2)))`

= `lim_(x→1) (x - 1)/((x - 1)(x^2 + x + 1)(sqrt(x + 3) + 2))`

= `lim_(x→1) 1/((x^2 + x + 1)(sqrt(x + 3) + 2)) ...[("As"  x→1","  x ≠ 1),(therefore x - 1 ≠ 0)]`

= `1/(lim_(x→1)(x^2 + x + 1) xx lim_{x→1} (sqrt(x + 3) + 2))`

= `1/((1^2 + 1 + 1) xx (sqrt(1 + 3) + 2))`

= `1/(3(2 + 2))`

= `1/12`

∴ `lim_(x→1) "f"(x) ≠ "f"(1)`

∴ f is discontinuous at x = 1