The limit of the given expression as $x \to 2$ is:

$\lim_{x \to 2} \frac{x^2 + 4x + 7}{x^2 - 3x + 1} = -19$

Would you like further details on how this result was obtained or have any questions?

Here are 5 related questions for further exploration:

1. How do you determine when you can apply direct substitution in limits?
2. What techniques are used when direct substitution leads to an indeterminate form like $\frac{0}{0}$?
3. How does L'Hôpital's rule help in solving limits with indeterminate forms?
4. Can all rational functions have limits at every point?
5. How does the behavior of the numerator and denominator affect the limit?

Tip: Always check for simplifications or factorization before concluding a limit result, especially for indeterminate forms like $\frac{0}{0}$.