https://file.aiquickdraw.com/mathbot/file/17255257281792h7d2rdc.jpg

The limit you uploaded is:

\[
\lim_{{x \to 6}} \frac{{x^2 - 12x + 36}}{{x^2 - 10x + 24}}
\]

Let's simplify the expression first.

### Step 1: Factor both the numerator and the denominator

- The numerator \( x^2 - 12x + 36 \) can be factored as:
  \[
  (x - 6)(x - 6) = (x - 6)^2
  \]

- The denominator \( x^2 - 10x + 24 \) can be factored as:
  \[
  (x - 6)(x - 4)
  \]

### Step 2: Rewrite the expression using the factored forms
\[
\frac{(x - 6)^2}{(x - 6)(x - 4)}
\]

### Step 3: Simplify the expression
Cancel out one factor of \( (x - 6) \) from the numerator and denominator (since \( x \neq 6 \) to avoid division by zero):
\[
\frac{x - 6}{x - 4}
\]

### Step 4: Evaluate the limit as \( x \to 6 \)
Substitute \( x = 6 \) into the simplified expression:
\[
\frac{6 - 6}{6 - 4} = \frac{0}{2} = 0
\]

Thus, the limit is:
\[
\boxed{0}
\]

Would you like a further breakdown, or have any other questions?

Here are 5 related questions for further practice:
1. What happens if the factor \(x - 6\) is in the denominator but cannot be canceled?
2. How do limits behave when there is no common factor in the numerator and denominator?
3. What is the limit of \(\frac{x^2 - 9x + 20}{x^2 - 8x + 15}\) as \(x \to 5\)?
4. How do you evaluate limits involving infinity, such as \(\lim_{{x \to \infty}} \frac{2x^2 + 3x + 1}{x^2 + 5x + 6}\)?
5. Can you explain the difference between one-sided limits and two-sided limits?

**Tip**: Always check if direct substitution yields an indeterminate form (like \( \frac{0}{0} \)) before simplifying expressions in limits.

lim (x -> 6) (x^2 - 12x + 36) / (x^2 - 10x + 24)

Solve the limit of (x^2 - 12x + 36) / (x^2 - 10x + 24) as x approaches 6 by factoring and simplifying the expression. This problem covers concepts of limits and algebraic simplification, suitable for students in Grades 10-12.

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