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 evaluate the limit:

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

First, let's analyze the form of the limit. When substituting \( x = 7 \) directly:

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

We get the indeterminate form \( \frac{0}{0} \). Since this is indeterminate, we need to simplify the expression.

### Step 1: Use Algebraic Manipulation (Rationalizing)
We can rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator:

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

This gives:

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

### Step 2: Simplify the Expression
The \( x - 49 \) terms in the numerator and denominator cancel out, leaving:

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

### Step 3: Evaluate the Limit
Now, substitute \( x = 7 \) into the simplified expression:

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

Thus, the value of the limit is:

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

### Would you like further clarification on this process, or any additional details?

Here are 5 related questions for further exploration:
1. How do you rationalize expressions with different forms of radicals?
2. Can limits always be solved by rationalizing the numerator?
3. What is the significance of the indeterminate form \( \frac{0}{0} \)?
4. How does L'Hôpital's Rule apply to limits like this one?
5. How would the limit change if we had \( \lim_{{x \to c}} \) where \( c \neq 7 \)?

**Tip:** Always check for an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) before proceeding with limit evaluation, as it indicates further algebraic manipulation is necessary.

Use the properties of limits to evaluate the limit if it exists: lim (x → 7) (sqrt(x) - 7)/(x - 49)

Learn how to evaluate the limit lim (x → 7) (sqrt(x) - 7)/(x - 49) using algebraic manipulation and rationalization. This problem involves resolving an indeterminate form and simplifying the expression step-by-step. Suitable for high school students in Grades 11-12.

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