Question

If function

f(x) = `x - |x|/x, x < 0`
      = `x + |x|/x, x > 0`
      = 1, x = 0, then

Options

• `lim_(x->0^-)`f(x) does not exist

• `lim_(x->0^+)`f(x) does not exist

• f(x) is continuous at x = 0

• `lim_(x->0^-) "f"(x) ne  lim_(x->0^+) "f"(x)`

Answer 1

f(x) is continuous at x = 0

Explanation:

Given function,

f(x) = `x - |x|/x, x < 0`
      = `x + |x|/x, x > 0 = 1,` x = 0

Now, at x = 0

LHL =`lim_(x->0^-) "f"(x) = lim_(x->0^-) (x - |x|/x)`

`lim_(x->0)(x - ((- x))/x) = lim_(x->0^-) (x + 1) = 1`

RHL = `lim_(x->0^+) "f"(x) = lim_(x->0^+) + (x + |x|/x)`

`lim_(x->0) (x + 1) = 1` and f(0) = 1

`therefore lim_(x->0^-) "f"(x) = lim_(x->0^+) "f"(x)` = f(0) = 1

⇒ f(x) is continuous at x = 0