Question

Find the limit: $ \lim_{x \to 0^+} 2 \left\lfloor x \right\rfloor - \frac{x}{|x|} $

• A. \( -2 \)
• B. \( 0 \)
• C. \( 2 \)
• D. Undefined

Hint:

When working with limits involving the floor function, recall that the floor function rounds down to the nearest integer. Additionally, for limits involving absolute values, remember that \( \frac{x}{|x|} \) simplifies based on the sign of \( x \).

Answer 1

The Correct Option is A

We are asked to find the limit of the following expression: \[ \lim_{x \to 0^+} 2 \left\lfloor x \right\rfloor - \frac{x}{|x|} \] As \( x \to 0^+ \), \( \left\lfloor x \right\rfloor \) will be 0 because \( \left\lfloor x \right\rfloor \) is the greatest integer less than or equal to \( x \), and for values of \( x \) between 0 and 1, \( \left\lfloor x \right\rfloor = 0 \). Therefore, the first term becomes: \[ 2 \left\lfloor x \right\rfloor = 2 \times 0 = 0 \] Now, for the second term, as \( x \to 0^+ \), \( \frac{x}{|x|} = 1 \) because \( |x| = x \) when \( x>0 \). So, the second term becomes: \[ \frac{x}{|x|} = 1 \]
Thus, the entire expression becomes: \[ 0 - 1 = -1 \] Therefore, the correct answer is (A) \( -2 \).