Question

$Lt_{x \to 0}(\frac{|x|}{x}+x+2)=$

• A. 0
• B. 1
• C. 2
• D. does not exist

Hint:

Limits involving $|x|/x$ at 0 generally result in DNE due to the jump discontinuity.

Answer 1

The Correct Option is D

Step 1: Concept A limit exists only if the Left Hand Limit (LHL) and Right Hand Limit (RHL) are equal.

Step 2: Meaning $|x|/x = 1$ for $x > 0$ and $|x|/x = -1$ for $x < 0$.

Step 3: Analysis RHL ($x \to 0^+$): $1 + 0 + 2 = 3$. LHL ($x \to 0^-$): $-1 + 0 + 2 = 1$.

Step 4: Conclusion Since LHL $\neq$ RHL, the limit does not exist. Final Answer: (D)