Question

If `f(x) = {{:(x + 2",", x < 0),(-x^2 - 2",", 0 ≤ x < 1),(x",", x ≥ 1):}` 

then the number of point(s) of discontinuity of f(x), is/are:

विकल्प

• 1

• 3

• 2

• 0

Answer 1

2

Explanation:

`f(x) = {{:(x + 2",", x < 0),(-x^2 - 2",", 0 ≤ x < 1),(x",", x ≥ 1):}`

Check continuity at x = 0

LHL = `lim_(h -> 0^-) f(0 - h)`

⇒ `lim_(h -> 0) -  h + 2 = 2`

RHL = `lim_(h -> 0) f(0 + h)`

⇒ `lim_(h -> 0) -  h^2 - 2 = -2`

LHL ≠ RHL

∴ f(x) is discontinuous at x = 0.

At x = 1.

LHL = `lim_(x -> 1^-) f(x)`

= `lim_(h -> 0) f(1 - h)`

= `lim_(h -> 0) - (1 - h)^2 - 2`

= –1 – 2

= –3

RHL = `lim_(x -> 1^+) f(x)`

= `lim_(h -> 0) f(1 + h)`

= `lim_(h -> 0) 1 + h`

= 1

Then, LHL ≠ RHL

∴ f(x) is discontinuous at x = 1.

The number of points of discontinuity is 2.