https://file.aiquickdraw.com/mathbot/file/1726616364958izmgdwxt.jpg

From the image, the expression seems to be:

\[
\lim_{{x \to y}} \frac{{x^2 - 16y}}{{x - 4y}}
\]

Now let's solve it step by step:

### Step 1: Simplify the numerator and denominator
The numerator is \( x^2 - 16y \), which is a difference of squares. We can rewrite it as:

\[
x^2 - 16y = (x - 4\sqrt{y})(x + 4\sqrt{y})
\]

### Step 2: Apply the limit as \( x \to 4y \)
Before plugging in \( x = 4y \), we first simplify the expression. Notice that the denominator is \( x - 4y \). So, we can cancel the common term \( (x - 4y) \) from both the numerator and denominator:

\[
\lim_{{x \to y}} \frac{{(x - 4y)(x + 4y)}}{{x - 4y}} = \lim_{{x \to y}} (x + 4y)
\]

### Step 3: Substitute \( x = 4y \)
Now, substitute \( x = 4y \) into the simplified expression:

\[
x + 4y = 4y + 4y = 8y
\]

So, the final answer is:

\[
\lim_{{x \to y}} \frac{{x^2 - 16y}}{{x - 4y}} = 8y
\]

Would you like more details on any part of the solution? Here are five related questions:

1. How do you factor more complex quadratic expressions?
2. What is the difference of squares rule?
3. How do you find limits involving indeterminate forms?
4. How do you deal with limits at infinity?
5. How can you verify limits using L'Hopital's Rule?

**Tip:** Always check if you can simplify expressions before applying the limit, especially for indeterminate forms.

Solve the limit problem \lim_{{x \to y}} \frac{{x^2 - 16y}}{{x - 4y}} step by step.

A detailed solution for the limit problem \lim_{{x \to y}} \frac{{x^2 - 16y}}{{x - 4y}}. Learn to apply the difference of squares rule and simplify expressions to solve limit problems. Suitable for students in Grades 10-12.

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