To solve the given limit problem:

$\lim_{x \to 5} \frac{x^2 - 21x + 15}{x^2 - 11x + 30}$

we need to simplify the expression and determine the limit as $x$ approaches 5. Let's go through the steps:

1. Factor the numerator and the denominator:

   • The numerator $x^2 - 21x + 15$ can be factored (if possible).
   • The denominator $x^2 - 11x + 30$ can be factored (if possible).

2. Simplify the fraction:

   Cancel any common factors between the numerator and the denominator.

3. Substitute $x = 5$ into the simplified expression:

   Evaluate the limit using the simplified expression.

Let's factor and simplify the expression:

Factoring:

• The numerator $x^2 - 21x + 15$ does not factor nicely with integer factors.
• The denominator $x^2 - 11x + 30$ factors as $(x - 5)(x - 6)$.

Since $x = 5$ is a root of the denominator, substituting directly would cause division by zero. So, let's factor and simplify:

$\frac{x^2 - 21x + 15}{x^2 - 11x + 30} = \frac{x^2 - 21x + 15}{(x - 5)(x - 6)}$

We observe the numerator does not share a factor of $(x - 5)$, and substituting $x = 5$ will still result in division by zero.

Thus, the limit does not exist because the expression has a discontinuity at $x = 5$. This occurs because, after factorization, there is no way to cancel the term causing the zero in the denominator.

If you have further questions or need additional details, feel free to ask!

Here are 5 related questions you might find helpful:

1. How do you factor quadratic equations in general?
2. What are the conditions under which a limit does not exist?
3. How do you simplify rational expressions involving polynomials?
4. What is the importance of checking continuity when evaluating limits?
5. How can you determine if a function is continuous at a point?

Tip: When solving limit problems involving rational expressions, always check for factors that can cancel out potential zeroes in the denominator to avoid undefined expressions.