Find the limit lim _(xarrow 0)(sqrt (x+5)-sqrt (5))/(x) (1)/(2sqrt (5)) infty (1)/(4) 1 DNE

## Step 1: Rationalizing the Numerator<br />### To evaluate the limit, we'll rationalize the numerator by multiplying the expression by $\frac{\sqrt{x+5} + \sqrt{5}}{\sqrt{x+5} + \sqrt{5}}$. This will help simplify the expression and eliminate the indeterminate form $\frac{0}{0}$ as $x$ approaches 0.<br /><br />## Step 2: Simplifying the Expression<br />### After multiplying, the numerator becomes $(x+5) - 5 = x$, and the denominator becomes $x(\sqrt{x+5} + \sqrt{5})$. We can then cancel out the $x$ terms in the numerator and denominator, leaving $\frac{1}{\sqrt{x+5} + \sqrt{5}}$.<br /><br />## Step 3: Evaluating the Limit<br />### Now, we can substitute $x=0$ into the simplified expression: $\frac{1}{\sqrt{0+5} + \sqrt{5}} = \frac{1}{2\sqrt{5}}$.