We are given the function:

$f(x) = \sqrt{x^2 + 13} - 7x - 6$

We are tasked with determining the limit $\lim_{x \to 6} f(x)$.

Step 1: Direct Substitution

First, let's try direct substitution of $x = 6$ into the function:

$f(6) = \sqrt{6^2 + 13} - 7(6) - 6$ $f(6) = \sqrt{36 + 13} - 42 - 6$ $f(6) = \sqrt{49} - 42 - 6$ $f(6) = 7 - 42 - 6$ $f(6) = -41$

Conclusion:

Since the direct substitution works without any indeterminate form, the limit is:

$\lim_{x \to 6} f(x) = -41$

Would you like more details or have any questions on this? Here are 5 related questions to deepen your understanding:

1. How do we handle cases when direct substitution results in an indeterminate form?
2. What are the typical techniques used to solve limits involving square roots?
3. How does the continuity of functions affect the calculation of limits?
4. What are some common pitfalls when calculating limits directly?
5. Can you apply limits to piecewise functions similarly?

Tip: When direct substitution gives a valid number, it is often the quickest way to find the limit, assuming the function is continuous at that point.