The image contains a limit expression involving square roots and polynomial terms. Let's extract and analyze the mathematical content from the image:

It appears to show:

$\lim_{x \to 2} \frac{\sqrt{x^3 + 5x} - \sqrt{4x^2 + 12x}}{x - 2}$

This is a limit problem, and it seems like the limit approaches $x = 2$. In its current form, directly substituting $x = 2$ leads to an indeterminate form of $\frac{0}{0}$, suggesting that we might need to simplify it further by rationalizing the numerator.

Solution:

To simplify the expression, we'll rationalize the numerator:

$\frac{\sqrt{x^3 + 5x} - \sqrt{4x^2 + 12x}}{x - 2}$

Multiply both the numerator and the denominator by the conjugate of the numerator:

$\frac{\sqrt{x^3 + 5x} - \sqrt{4x^2 + 12x}}{x - 2} \times \frac{\sqrt{x^3 + 5x} + \sqrt{4x^2 + 12x}}{\sqrt{x^3 + 5x} + \sqrt{4x^2 + 12x}}$

This simplifies to:

$\frac{(\sqrt{x^3 + 5x})^2 - (\sqrt{4x^2 + 12x})^2}{(x - 2)(\sqrt{x^3 + 5x} + \sqrt{4x^2 + 12x})}$

Now simplify the numerator:

$(x^3 + 5x) - (4x^2 + 12x)$

This becomes:

$x^3 - 4x^2 + 5x - 12x = x^3 - 4x^2 - 7x$

So the expression is now:

$\frac{x^3 - 4x^2 - 7x}{(x - 2)(\sqrt{x^3 + 5x} + \sqrt{4x^2 + 12x})}$

At this stage, we can attempt to factor $x^3 - 4x^2 - 7x$ to check for any simplifications that could help resolve the limit.

Would you like to continue with factoring or explore an alternative method for solving this limit?

Here are some related questions to expand your understanding:

1. What is the method of rationalizing the numerator, and why is it useful in limit problems?
2. How do you factor cubic expressions like $x^3 - 4x^2 - 7x$?
3. What are indeterminate forms in limits, and how do they guide us in choosing methods like L'Hopital's Rule or rationalization?
4. Could L'Hopital's Rule be applied here instead of rationalizing?
5. How can you check whether a limit exists when dealing with complicated square root expressions?

Tip: Rationalizing is often used to simplify expressions involving square roots in limit problems to eliminate indeterminate forms.