We are given the limit:

$\lim_{x \to 2} \left( a\sqrt{x+7} + \frac{b}{x-2} \right) = 3$

We need to find the value of $3a + b$.

Step 1: Substitute $x = 2$ into the expression

First, substitute $x = 2$ into the expression:

$a\sqrt{2+7} + \frac{b}{2-2}$

This simplifies to:

$a\sqrt{9} + \frac{b}{0} = 3a + \frac{b}{0}$

Since $\frac{b}{0}$ is undefined unless $b = 0$, we conclude $b = 0$ to avoid an undefined expression.

Step 2: Substitute $b = 0$ into the limit

Substituting $b = 0$ into the limit expression, we have:

$\lim_{x \to 2} \left( a\sqrt{x+7} \right) = 3$

Now, substitute $x = 2$ into the simplified expression:

$a\sqrt{9} = 3$

This simplifies to:

$3a = 3$

So, $a = 1$.

Step 3: Calculate $3a + b$

With $a = 1$ and $b = 0$:

$3a + b = 3(1) + 0 = 3$

Thus, $3a + b = 3$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the solution change if $b$ were not zero?
2. What if the limit had been $\lim_{x \to 2} \left( a\sqrt{x+7} + \frac{b}{x-2} \right) = 0$?
3. Can we find $a$ and $b$ if we know other conditions on the limit?
4. How does the concept of limits help in understanding the behavior of functions near undefined points?
5. What would happen if the function inside the limit were more complex, such as involving higher powers or trigonometric functions?

Tip: When evaluating limits, always check if the function results in an indeterminate form to guide your approach in solving.