https://file.aiquickdraw.com/mathbot/file/17255321623163sj98075.jpg

The image shows a mathematical expression involving a limit:

\[
\lim_{{x \to 10}} \frac{25 - x^2}{x - 10}
\]

To solve this limit, notice that directly substituting \(x = 10\) would lead to a division by zero. We should attempt to simplify the expression by factoring.

### Step-by-Step Solution:

1. **Rewrite the numerator:**
   The numerator \(25 - x^2\) is a difference of squares and can be factored as:
   \[
   25 - x^2 = (5 - x)(5 + x)
   \]

2. **Simplify the expression:**
   Now the expression becomes:
   \[
   \lim_{{x \to 10}} \frac{(5 - x)(5 + x)}{x - 10}
   \]
   Notice that \(5 - x = -(x - 5)\), so:
   \[
   \lim_{{x \to 10}} \frac{-(x - 5)(x + 5)}{x - 10}
   \]
   Also, \(x - 10 = -(10 - x)\), which gives us:
   \[
   \lim_{{x \to 10}} \frac{-(x - 5)(x + 5)}{-(10 - x)}
   \]

3. **Simplify further:**
   After canceling out the negative signs:
   \[
   \lim_{{x \to 10}} \frac{(x - 5)(x + 5)}{10 - x}
   \]

4. **Substitute the value:**
   Now substitute \(x = 10\):
   \[
   \frac{(10 - 5)(10 + 5)}{10 - x}
   \]

Hence:
\[
=0

Any other questions?

Learn how to solve the limit lim (x -> 10) (25 - x^2) / (x - 10) using factoring techniques. This problem involves recognizing the difference of squares and simplifying the expression. Suitable for high school students, Grades 10-12.

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