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The given expression is:

\[
\lim_{{x \to 0}} \frac{{\sqrt{49 - x} - 7}}{x}
\]

### Step-by-step solution:

This is an indeterminate form of \(\frac{0}{0}\) when directly substituting \(x = 0\). So, we need to manipulate the expression to resolve this.

#### 1. Multiply by the conjugate:
We can multiply the numerator and denominator by the conjugate of the numerator to simplify the square root term. The conjugate of \(\sqrt{49 - x} - 7\) is \(\sqrt{49 - x} + 7\).

\[
\lim_{{x \to 0}} \frac{{\sqrt{49 - x} - 7}}{x} \times \frac{{\sqrt{49 - x} + 7}}{{\sqrt{49 - x} + 7}} = \lim_{{x \to 0}} \frac{{(\sqrt{49 - x} - 7)(\sqrt{49 - x} + 7)}}{{x(\sqrt{49 - x} + 7)}}
\]

#### 2. Simplify the numerator:
By applying the difference of squares formula \((a - b)(a + b) = a^2 - b^2\), we simplify the numerator:

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

Thus, the expression becomes:

\[
\lim_{{x \to 0}} \frac{{-x}}{{x(\sqrt{49 - x} + 7)}} = \lim_{{x \to 0}} \frac{{-1}}{{\sqrt{49 - x} + 7}}
\]

#### 3. Evaluate the limit:
Now, substitute \(x = 0\) into the expression:

\[
\frac{{-1}}{{\sqrt{49 - 0} + 7}} = \frac{{-1}}{{7 + 7}} = \frac{{-1}}{{14}}
\]

### Final Answer:
\[
\lim_{{x \to 0}} \frac{{\sqrt{49 - x} - 7}}{x} = \frac{{-1}}{{14}}
\]

Would you like further explanation on any step or more examples of limits?

Here are 5 related questions for further practice:
1. How would you approach solving \(\lim_{{x \to 0}} \frac{{\sqrt{x+1} - 1}}{x} \)?
2. What is the limit of \(\lim_{{x \to 0}} \frac{{1 - \cos(x)}}{x^2} \)?
3. Can you compute \(\lim_{{x \to 0}} \frac{{\sin(3x)}}{x}\)?
4. How do we apply L'Hopital's Rule to solve limits of indeterminate forms?
5. What are the different techniques for evaluating limits involving radicals?

**Tip:** Always check if a limit expression is in an indeterminate form before applying algebraic manipulations or L'Hopital's Rule!

Learn how to evaluate the limit lim (x → 0) (sqrt(49 - x) - 7) / x by multiplying by the conjugate. This problem involves simplifying an indeterminate form using algebraic techniques. Suitable for students in Grades 11-12.

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