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The image contains the following limit problems:

1. \(\lim_{{x \to 10}} \frac{{x - 10}}{{x - 10}}\)
2. \(\lim_{{x \to -30}} \frac{{36 - x^2}}{{x^2 - 30}}\)

### Problem 1: 
\[
\lim_{{x \to 10}} \frac{{x - 10}}{{x - 10}}
\]
This is a standard limit problem where the numerator and denominator are identical. We can simplify the expression:
\[
\frac{{x - 10}}{{x - 10}} = 1 \quad \text{(for all } x \neq 10\text{)}.
\]
Thus, the limit becomes:
\[
\lim_{{x \to 10}} 1 = 1
\]
So, the limit is **1**.

### Problem 2:
\[
\lim_{{x \to -30}} \frac{{36 - x^2}}{{x^2 - 30}}
\]
We need to substitute \(x = -30\) into the expression. If we do that directly, we get:
\[
36 - (-30)^2 = 36 - 900 = -864
\]
and
\[
(-30)^2 - 30 = 900 - 30 = 870.
\]
Thus, the expression becomes:
\[
\frac{{-864}}{{870}}.
\]
Simplifying this fraction:
\[
\frac{{-864}}{{870}} = -\frac{{864}}{{870}} = -\frac{{432}}{{435}}.
\]
This is the simplified form of the limit, which equals approximately \(-0.9931\).

Would you like further details on any steps or calculations?

### Related questions:
1. What are the conditions for limits to exist?
2. How do we deal with indeterminate forms when calculating limits?
3. What is the concept of one-sided limits?
4. How can L'Hopital's Rule be applied to limits of indeterminate forms?
5. How does simplifying rational functions help in solving limits?

### Tip:
Always check if direct substitution gives a determinate value or an indeterminate form like \(\frac{0}{0}\). If indeterminate, explore methods like factoring, rationalization, or L'Hopital's Rule.

Evaluate the following limits:
1. lim (x → 10) (x - 10)/(x - 10)
2. lim (x → -30) (36 - x^2)/(x^2 - 30)

This guide walks through the process of evaluating limits involving rational functions with indeterminate forms. Learn how to solve problems such as lim (x → 10) (x - 10)/(x - 10) and lim (x → -30) (36 - x^2)/(x^2 - 30), using limit laws and simplification techniques. Suitable for Grades 11-12.

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