Question

$\lim_{x \to 0} \frac{|x|}{|x| + x^2} =$}

• A. 0
• B. 1
• C. -1
• D. $\frac{1}{2}$

Hint:

Since $x^2 = |x|^2$, you can rewrite the expression as $|x|/(|x| + |x|^2)$ and cancel $|x|$.

Answer 1

The Correct Option is B

Step 1: Concept
Evaluate the limit by considering $x \to 0^+$ and $x \to 0^-$ due to the absolute value.

Step 2: Meaning
For $x \ne 0$, we can divide the numerator and denominator by $|x|$.

Step 3: Analysis
$\frac{|x|}{|x| + x^2} = \frac{|x|}{|x|(1 + \frac{x^2}{|x|})} = \frac{1}{1 + |x|}$.

Step 4: Conclusion
As $x \to 0$, $1 + |x| \to 1$. The limit is $1/1 = 1$. Final Answer: (B)