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Let's solve both parts of the problem to find the given limits.

### a) \(\lim_{{x \to 10}} \frac{{x^2 - 15x + 50}}{{(x - 10)^2}}\)

To solve this limit, we first need to simplify the expression in the numerator:

\[
x^2 - 15x + 50
\]

We can factorize the quadratic expression:

\[
x^2 - 15x + 50 = (x - 10)(x - 5)
\]

So, the limit becomes:

\[
\lim_{{x \to 10}} \frac{{(x - 10)(x - 5)}}{{(x - 10)^2}}
\]

We can cancel out one \((x - 10)\) term from the numerator and the denominator:

\[
\lim_{{x \to 10}} \frac{{x - 5}}{{x - 10}}
\]

Now, substitute \(x = 10\):

\[
\frac{{10 - 5}}{{10 - 10}} = \frac{5}{0}
\]

This indicates that the limit approaches infinity (since the denominator goes to 0 while the numerator remains a constant). Therefore, the limit does not exist in the usual sense; it diverges to infinity.

### b) \(\lim_{{x \to 3}} \frac{{x + 3}}{{x - 3}}\)

Substitute \(x = 3\):

\[
\frac{{3 + 3}}{{3 - 3}} = \frac{6}{0}
\]

This is also a division by zero situation. To understand the behavior, we need to determine the limit as \(x\) approaches 3 from both sides (left and right).

1. As \(x \to 3^+\) (from the right), \((x - 3)\) is a small positive number, so the fraction \(\frac{x+3}{x-3} \to +\infty\).
2. As \(x \to 3^-\) (from the left), \((x - 3)\) is a small negative number, so the fraction \(\frac{x+3}{x-3} \to -\infty\).

Since the left-hand and right-hand limits are not equal, the limit does not exist.

### Final Answers:
- a) \(\lim_{{x \to 10}} \frac{{x^2 - 15x + 50}}{{(x - 10)^2}} = +\infty\)
- b) \(\lim_{{x \to 3}} \frac{{x + 3}}{{x - 3}}\) does not exist.

Would you like a deeper explanation or have any questions?

### Related Questions to Explore:
1. What are the different types of discontinuities that can occur when evaluating limits?
2. How do we use L'Hopital's rule to solve limits involving indeterminate forms?
3. How can factoring help in simplifying limits involving rational functions?
4. What does it mean for a limit to diverge to infinity, and how is

Find the limit of the following expressions:
 a) lim (x → 10) [(x^2 - 15x + 50) / (x - 10)^2]
 b) lim (x → 3) [(x + 3) / (x - 3)]

Solve the limit problems involving rational functions and infinite limits. The first problem requires factoring a quadratic expression, while the second investigates the behavior of a rational function as the denominator approaches zero. Suitable for students in Grades 10-12.

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