The given limit expression is:

\[
\lim_{x \to 13} \frac{\sqrt{x + 3} - 4}{3x - 39}
\]

Let's solve this step by step.

### Step 1: Substitute \(x = 13\) into the expression
Substituting \(x = 13\):

\[
\frac{\sqrt{13 + 3} - 4}{3(13) - 39} = \frac{\sqrt{16} - 4}{39 - 39} = \frac{4 - 4}{0} = \frac{0}{0}
\]

This results in an indeterminate form \(\frac{0}{0}\). Therefore, we need to simplify the expression further by using algebraic techniques.

### Step 2: Rationalize the numerator
To resolve the indeterminate form, we rationalize the numerator by multiplying the numerator and denominator by the conjugate of the numerator:

\[
\text{Multiply by } \frac{\sqrt{x + 3} + 4}{\sqrt{x + 3} + 4}:
\]

\[
\frac{\sqrt{x + 3} - 4}{3x - 39} \cdot \frac{\sqrt{x + 3} + 4}{\sqrt{x + 3} + 4} = \frac{(\sqrt{x + 3})^2 - 4^2}{(3x - 39)(\sqrt{x + 3} + 4)}
\]

### Step 3: Simplify the numerator
Simplifying the numerator:

\[
(\sqrt{x + 3})^2 - 4^2 = (x + 3) - 16 = x - 13
\]

Now, the expression becomes:

\[
\frac{x - 13}{(3x - 39)(\sqrt{x + 3} + 4)}
\]

### Step 4: Simplify the denominator
Notice that \(3x - 39 = 3(x - 13)\), so the expression becomes:

\[
\frac{x - 13}{3(x - 13)(\sqrt{x + 3} + 4)}
\]

### Step 5: Cancel the common factor
We can cancel the common factor of \(x - 13\) from the numerator and denominator:

\[
\frac{1}{3(\sqrt{x + 3} + 4)}
\]

### Step 6: Substitute \(x = 13\) into the simplified expression
Now, substitute \(x = 13\) into the simplified expression:

\[
\frac{1}{3(\sqrt{13 + 3} + 4)} = \frac{1}{3(4 + 4)} = \frac{1}{3 \times 8} = \frac{1}{24}
\]

### Final Answer:
The limit is:

\[
\lim_{x \to 13} \frac{\sqrt{x + 3} - 4}{3x - 39} = \frac{1}{24}
\]

Would you like any additional details or clarifications on this? 

Here are 5 related questions for further practice:

1. Evaluate \(\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}\).
2. Solve \(\lim_{x \to 5} \frac{\sqrt{x + 1} - 3}{x - 8}\).
3. Calculate \(\lim_{x \to 0} \frac{x^2 - 4x}{x^2 - 2x}\).
4. Determine \(\lim_{x \to 0} \frac{x + 1 - 1}{x^2}\).
5. Find \(\lim_{x \to 16} \frac{\sqrt{x} - 4}{x - 16}\).

**Tip**: Always check for indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) before simplifying limits.

Learn how to solve the limit lim x→13 (sqrt(x+3)-4)/(3x-39) by rationalizing the numerator to eliminate the indeterminate form 0/0. Follow the step-by-step solution to simplify the expression and find the limit, which results in 1/24. Suitable for Grades 10-12.

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