The limit problem from the image is:

$\lim_{x \to 3} \frac{\sqrt{x^2 - 5} - 2}{2x - 6}$

Step 1: Simplify the expression

Notice that when you directly substitute $x = 3$ into the expression:

• The denominator becomes $2(3) - 6 = 0$
• The numerator becomes $\sqrt{3^2 - 5} - 2 = \sqrt{9 - 5} - 2 = \sqrt{4} - 2 = 2 - 2 = 0$

Thus, direct substitution leads to the indeterminate form $\frac{0}{0}$.

Step 2: Apply algebraic manipulation

We will rationalize the numerator to eliminate the square root. Multiply both the numerator and denominator by the conjugate of the numerator:

$\frac{\sqrt{x^2 - 5} - 2}{2x - 6} \times \frac{\sqrt{x^2 - 5} + 2}{\sqrt{x^2 - 5} + 2}$

The numerator becomes:

$(\sqrt{x^2 - 5} - 2)(\sqrt{x^2 - 5} + 2) = (x^2 - 5) - 4 = x^2 - 9$

The denominator becomes:

$(2x - 6)(\sqrt{x^2 - 5} + 2) = (2(x - 3))(\sqrt{x^2 - 5} + 2)$

Thus, the expression simplifies to:

$\frac{x^2 - 9}{2(x - 3)(\sqrt{x^2 - 5} + 2)}$

Step 3: Factor and cancel terms

We can factor the numerator $x^2 - 9$ as:

$x^2 - 9 = (x - 3)(x + 3)$

Now, cancel the common factor of $x - 3$ from the numerator and denominator:

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

Step 4: Substitute $x = 3$

Now that the indeterminate form has been removed, substitute $x = 3$:

$\frac{3 + 3}{2(\sqrt{3^2 - 5} + 2)} = \frac{6}{2(\sqrt{9 - 5} + 2)} = \frac{6}{2(\sqrt{4} + 2)} = \frac{6}{2(2 + 2)} = \frac{6}{8} = \frac{3}{4}$

Final Answer:

The limit is $\frac{3}{4}$, so the correct choice is B. 3/4.

Would you like further details or have any questions?

Here are some related questions for practice:

1. How do you rationalize a numerator involving a square root?
2. Why does direct substitution result in an indeterminate form?
3. What is the general method to deal with indeterminate forms in limits?
4. How does canceling common factors help simplify limits?
5. What other techniques are useful for solving limits involving square roots?

Tip: Rationalizing the numerator is a useful technique when you encounter limits involving square roots that lead to indeterminate forms.