We are asked to find the limit:

$\lim_{x \to 5} \frac{x^2 - 7x + 10}{x - 5}$

Step-by-step solution:

1. Factor the numerator: We can try factoring the quadratic expression in the numerator, $x^2 - 7x + 10$. Factoring gives: $x^2 - 7x + 10 = (x - 5)(x - 2)$

2. Simplify the expression: The limit expression now becomes: $\frac{(x - 5)(x - 2)}{x - 5}$ Since $x \neq 5$, we can cancel out the common factor of $x - 5$, leaving: $\lim_{x \to 5} (x - 2)$

3. Evaluate the limit: Now, we substitute $x = 5$ into the simplified expression: $5 - 2 = 3$

Thus, the value of the limit is 3.

Let me know if you would like more details on this or have any questions.

Related questions:

1. What is the general technique to factor a quadratic equation?
2. How do we deal with indeterminate forms in limits?
3. What is L'Hôpital's Rule, and when would it apply here?
4. Can you explain how to handle limits involving trigonometric functions?
5. What if the denominator didn’t have a removable discontinuity? How would the solution change?

Tip:

Always check if you can factor expressions when facing limits that result in indeterminate forms, such as $\frac{0}{0}$, as simplifying may allow direct substitution.