Evaluate \displaystyle \lim_{x \to 0} \frac{\sqrt{2}x}{\sqrt{2+x}-\sqrt{2}}.
To evaluate the given limit, we can use algebraic manipulation to simplify the expression in the limit. Here's how we do it:
Step 1: Start with the given expression,
\[\frac{\sqrt{2}x}{\sqrt{2+x}-\sqrt{2}}.\]
Step 2: Rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{2+x}+\sqrt{2}\). This eliminates the square root terms in the denominator.
\[\frac{\sqrt{2}x}{\sqrt{2+x}-\sqrt{2}} \cdot \frac{\sqrt{2+x}+\sqrt{2}}{\sqrt{2+x}+\sqrt{2}} = \frac{\sqrt{2}x(\sqrt{2+x}+\sqrt{2})}{(\sqrt{2+x}-\sqrt{2})(\sqrt{2+x}+\sqrt{2})}.\]
Step 3: Simplify the expression further by distributing \(\sqrt{2}x\) into the numerator.
\[\frac{\sqrt{2}x\sqrt{2+x}+\sqrt{2}x\sqrt{2}}{(\sqrt{2+x})^2 - (\sqrt{2})^2} = \frac{\sqrt{2}x\sqrt{2+x}+2x}{2+x-2}.\]
Step 4: Simplify the numerator by factoring out \(x\).
\[x\left(\sqrt{2}\sqrt{2+x}+2\right).\]
Step 5: Cancel out the common factor of \(\sqrt{2}\) between the numerator and denominator.
\[x\left(\sqrt{2+x}+2\right).\]
Step 6: Finally, take the limit as \(x\) approaches \(0\) by plugging in \(0\) into the simplified expression.
\[\lim_{x \to 0} x(\sqrt{2+x}+2) = 0(\sqrt{2+0}+2) = 0 \cdot (2) = 0.\]
Therefore, the value of the given limit is \(0\).