Use the properties of limits to evaluate the limit if it exists.

lim is over x→7 (square root of (x) - 7/x - 49)

To solve the limit:

\[
\lim_{{x \to 7}} \frac{\sqrt{x} - 7}{x - 49}
\]

### Step 1: Direct Substitution

If we directly substitute \(x = 7\) into the expression:

\[
\frac{\sqrt{7} - 7}{7 - 49} = \frac{7 - 7}{7 - 49} = \frac{0}{0}
\]

We get the indeterminate form \( \frac{0}{0} \), so we need to apply algebraic techniques to resolve this.

### Step 2: Simplifying the Expression

We can simplify the expression by using the **conjugate** technique. Multiply both the numerator and the denominator by the conjugate of the numerator, which is \( \sqrt{x} + 7 \):

\[
\frac{\sqrt{x} - 7}{x - 49} \times \frac{\sqrt{x} + 7}{\sqrt{x} + 7}
\]

This simplifies the numerator:

\[
(\sqrt{x} - 7)(\sqrt{x} + 7) = (\sqrt{x})^2 - 7^2 = x - 49
\]

So the expression becomes:

\[
\frac{x - 49}{(x - 49)(\sqrt{x} + 7)}
\]

### Step 3: Cancel the Common Terms

We can cancel the common factor \(x - 49\) from both the numerator and denominator, provided \(x \neq 49\):

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

### Step 4: Substituting \(x = 7\)

Now substitute \(x = 7\) into the simplified expression:

\[
\frac{1}{\sqrt{7} + 7} = \frac{1}{7 + 7} = \frac{1}{14}
\]

### Final Answer:
\[
\lim_{{x \to 7}} \frac{\sqrt{x} - 7}{x - 49} = \frac{1}{14}
\]

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How do you apply the conjugate technique to rationalize expressions?
2. What are indeterminate forms, and why do they require special techniques?
3. Can L'Hopital's Rule be used for this limit? If so, how?
4. What is the difference between direct substitution and algebraic manipulation when finding limits?
5. How does the chain rule apply to limits involving square roots?

**Tip:** Always check if you can simplify a limit expression before jumping into advanced techniques like L'Hopital’s Rule.

Use the properties of limits to evaluate the limit if it exists. lim is over x→7 (square root of (x) - 7 / (x - 49))

Learn how to evaluate the limit lim (x→7) (sqrt(x) - 7) / (x - 49) using algebraic simplification and conjugates. This detailed solution demonstrates how to resolve an indeterminate form and apply limit cancellation rules, suitable for high school calculus students (Grades 11-12).

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