https://file.aiquickdraw.com/mathbot/file/1726773603227uttudb6h.jpg

The expression you uploaded is:

\[
\lim_{x \to 7} \frac{\sqrt{x} - \sqrt{7}}{x - 7}
\]

This is an indeterminate form (0/0) as \( x \to 7 \), so we can use a method to simplify the expression. One common method in such cases is multiplying by the conjugate to eliminate the square roots in the numerator.

Here are the steps to solve this limit:

### Step 1: Multiply by the conjugate
We multiply both the numerator and the denominator by the conjugate of the numerator, \( \sqrt{x} + \sqrt{7} \).

\[
\frac{\sqrt{x} - \sqrt{7}}{x - 7} \times \frac{\sqrt{x} + \sqrt{7}}{\sqrt{x} + \sqrt{7}} = \frac{(\sqrt{x} - \sqrt{7})(\sqrt{x} + \sqrt{7})}{(x - 7)(\sqrt{x} + \sqrt{7})}
\]

### Step 2: Simplify the numerator
Using the difference of squares formula, the numerator becomes:

\[
(\sqrt{x})^2 - (\sqrt{7})^2 = x - 7
\]

So the expression simplifies to:

\[
\frac{x - 7}{(x - 7)(\sqrt{x} + \sqrt{7})}
\]

### Step 3: Cancel the common factor
We can cancel out the \( x - 7 \) term from both the numerator and denominator (note that we can do this because we are taking the limit as \( x \to 7 \), not at \( x = 7 \)):

\[
\frac{1}{\sqrt{x} + \sqrt{7}}
\]

### Step 4: Apply the limit
Now we can directly substitute \( x = 7 \) into the remaining expression:

\[
\lim_{x \to 7} \frac{1}{\sqrt{x} + \sqrt{7}} = \frac{1}{\sqrt{7} + \sqrt{7}} = \frac{1}{2\sqrt{7}}
\]

### Final Answer:
The limit is:

\[
\frac{1}{2\sqrt{7}}
\]

Would you like further details or have any questions?

### Related Questions:
1. What is the general technique to solve limits involving square roots?
2. How does the difference of squares formula simplify such expressions?
3. Can the L'Hopital's Rule be applied to solve this type of limit problem?
4. Why is it valid to cancel out terms in limits, even when they might be zero?
5. How does the conjugate technique help in simplifying indeterminate forms?

**Tip:** Whenever square roots lead to indeterminate forms in limits, try multiplying by the conjugate to simplify the expression.

lim (x -> 7) (sqrt(x) - sqrt(7)) / (x - 7)

Learn how to solve the limit lim (x -> 7) (sqrt(x) - sqrt(7)) / (x - 7) using the conjugate method to eliminate the square root, simplifying the expression and applying the limit. Suitable for advanced math students.

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