The given problem involves evaluating the following limit:

$\lim_{{x \to 0}} \frac{\sqrt{2x + 3} - \sqrt{3}}{x}$

This limit can be interpreted as the derivative of a function at a certain point using the definition of a derivative. Specifically, it resembles the derivative of a square root function evaluated at a point, which we can identify and compute step by step.

Step 1: Recognize the Structure

The limit looks like the difference quotient form of a derivative, which is:

$f'(a) = \lim_{{x \to a}} \frac{f(x) - f(a)}{x - a}$

Comparing this to the limit we are given, it suggests that the function $f(x)$ is $f(x) = \sqrt{2x + 3}$, and the point $a = 0$ because $\sqrt{3}$ appears as the function evaluated at $x = 0$.

Step 2: Set up the Derivative

We now want to calculate $f'(x)$ for the function $f(x) = \sqrt{2x + 3}$, and then evaluate it at $x = 0$.

Using the chain rule for derivatives:

$f'(x) = \frac{d}{dx} \left( \sqrt{2x + 3} \right)$

First, differentiate $\sqrt{u}$ where $u = 2x + 3$:

$f'(x) = \frac{1}{2\sqrt{2x + 3}} \cdot \frac{d}{dx}(2x + 3)$

$f'(x) = \frac{1}{2\sqrt{2x + 3}} \cdot 2 = \frac{1}{\sqrt{2x + 3}}$

Step 3: Evaluate the Derivative at $x = 0$

Now, substitute $x = 0$ into the derivative:

$f'(0) = \frac{1}{\sqrt{2(0) + 3}} = \frac{1}{\sqrt{3}}$

Thus, the derivative of $f(x) = \sqrt{2x + 3}$ at $x = 0$ is $\frac{1}{\sqrt{3}}$.

Final Answer

The function $f(x) = \sqrt{2x + 3}$ and the derivative at $x = 0$ is $\frac{1}{\sqrt{3}}$.

Do you want more details or have any questions?

Here are five related questions that expand on this concept:

1. How would you compute the derivative if the function were $f(x) = \sqrt{4x + 1}$?
2. What happens if you substitute a different value for $x$ instead of 0 in the derivative?
3. How can L'Hopital's Rule be applied to evaluate limits like this one?
4. What is the geometrical interpretation of the derivative in this context?
5. How would the result change if the function involved a cube root instead of a square root?

Tip: When solving limits that resemble the derivative definition, always check if the limit matches the form of a difference quotient.