Question

If f(x) = `{{:((sqrt(1 - "ax") - sqrt(1 + "ax"))/"x"","   -1 ≤ "x" ≤ 0), ((3x + 1)/(x - 2)","   0 ≤ x ≤ 1):}` is continuous in [-1, 1], then a is equal to ______

Options

• -1

• `(-1)/2`

• `1/2`

• 1

Answer 1

If f(x) = `{{:((sqrt(1 - "ax") - sqrt(1 + "ax"))/"x"","   -1 ≤ "x" ≤ 0), ((3x + 1)/(x - 2)","   0 ≤ x ≤ 1):}` is continuous in [-1, 1], then a is equal to `underline(1/2)`.

Explanation:

Since, f(x) is continuous in [-1, 1].

∴ it is continuous at x = 0.

∴ `lim_{x→0^-} f(x) = lim_{x→0^+} f(x)`

⇒ `lim_{x→0} (sqrt(1 - ax) - sqrt(1 + ax))/x =  lim_{x→0} (3x + 1)/(x - 2)` 

⇒ `lim_{x→0} ((1 - ax) - (1 + ax))/(x(sqrt(1 - ax) + sqrt(1 + ax))) = (-1)/2`

⇒ a = `1/2`