Determine the limit of the function as h approaches 0: $\lim _{h \rightarrow 0} \frac{\sqrt{(2+h)^{2}-1}-\sqrt{3}}{h}$

Step 1 ：We are given the function \(\frac{\sqrt{(2+h)^{2}-1}-\sqrt{3}}{h}\) and we are asked to find the limit as h approaches 0.

Step 2 ：Since the function is not defined at h=0, we can't simply substitute h=0 into the function. This suggests that we should use L'Hopital's rule.

Step 3 ：L'Hopital's rule states that if the limit of a function as x approaches a certain value is of the form 0/0 or ∞/∞, then this limit is equal to the limit of the derivative of the numerator divided by the derivative of the denominator as x approaches the same value.

Step 4 ：First, we find the derivative of the numerator, which is \(\frac{(h + 2)}{\sqrt{(h + 2)^{2} - 1}}\).

Step 5 ：Next, we find the derivative of the denominator, which is 1.

Step 6 ：Finally, we find the limit of the ratio of these derivatives as h approaches 0, which is \(\frac{2\sqrt{3}}{3}\).

Step 7 ：Final Answer: The limit of the function as h approaches 0 is \(\boxed{\frac{2\sqrt{3}}{3}}\).