Question

If f(x) = `{((x^4 - 1/81)/(x^3 - 1/27), x ≠ 1/3), (k, x = 1/3):}` is continuous at x = `1/3`, then the value of k is ______

Options

• `4/3`

• `1/3`

• `4/9`

• `1/5`

Answer 1

If f(x) = `{((x^4 - 1/81)/(x^3 - 1/27), x ≠ 1/3), (k, x = 1/3):}` is continuous at x = `1/3`, then the value of k is `underline(4/9)`.

Explanation:

Since, f(x) is continuous at x = `1/3`.

∴ `f(1/3) = lim_{x→1/3}f(x)`

⇒ k = `lim_{x→1/3} (x^4 - 1/81)/(x^3 - 1/27)`

Applying L'Hospital rule on R.H.S., we get

k = `lim_{x→1/3} (4x^3)/(3x^2) = lim_{x→1/3} 4/3x = 4/3(1/3) = 4/9`