Question

If \[ f(x) = \frac{x + |x|}{x} \] then the value of \[ \lim_{x \to 0} f(x) \] is:

• A. 0
• B. 2
• C. Does not exist
• D. None of these

Hint:

For limits involving absolute values, always evaluate left-hand and right-hand limits separately.

Answer 1

The Correct Option is C

Step 1: Evaluating left-hand limit (LHL).
For \( x<0 \), we have \( |x| = -x \), so \[ f(x) = \frac{x + (-x)}{x} = \frac{0}{x} = 0 \] Thus, \[ \lim_{x \to 0^-} f(x) = 0 \] Step 2: Evaluating right-hand limit (RHL).
For \( x>0 \), we have \( |x| = x \), so \[ f(x) = \frac{x + x}{x} = \frac{2x}{x} = 2 \] Thus, \[ \lim_{x \to 0^+} f(x) = 2 \] Step 3: Checking if the limit exists.
Since \( \lim_{x \to 0^-} f(x) = 0 \) and \( \lim_{x \to 0^+} f(x) = 2 \), \[ \lim_{x \to 0} f(x) { does not exist} \] Thus, the correct answer is (C).