We are tasked with evaluating the following function's limits:

$f(x) = x \sqrt{2 + x^2}$

at three points:

• (a) $\lim\limits_{x \to 0^-} f(x)$
• (b) $\lim\limits_{x \to 0^+} f(x)$
• (c) $\lim\limits_{x \to 0} f(x)$

Solution Approach:

1. (a) Limit as $x \to 0^-$:

   • The function is continuous, and as $x$ approaches 0 from the left ($x \to 0^-$), the behavior of the square root term does not change because it involves $x^2$, which is non-negative.
   • Thus, we can directly substitute $x = 0$ into the function.

   [ \lim_{x \to 0^-} x \sqrt{2 + x^2} = 0 \cdot \sqrt{2 + 0^