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The given limit is:

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

To evaluate this limit, we can use the technique of multiplying by the conjugate to simplify the expression. Let's rewrite the expression:

\[
\lim_{{h \to 0}} \frac{\sqrt{49 + h} - 7}{h} \cdot \frac{\sqrt{49 + h} + 7}{\sqrt{49 + h} + 7}
\]

Now, the numerator becomes a difference of squares:

\[
= \lim_{{h \to 0}} \frac{(\sqrt{49 + h})^2 - 7^2}{h (\sqrt{49 + h} + 7)}
\]

Simplify the squares:

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

This reduces to:

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

Cancel the \(h\) terms:

\[
= \lim_{{h \to 0}} \frac{1}{\sqrt{49 + h} + 7}
\]

Now, as \(h \to 0\), we substitute \(h = 0\):

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

Thus, the value of the limit is:

\[
\boxed{\frac{1}{14}}
\]

Would you like further details on this process or have any questions?

Here are some related questions for further exploration:
1. What is the technique of using conjugates, and how is it applied in other limits?
2. How do you evaluate limits that involve radicals without using the conjugate method?
3. What is L'Hopital's rule, and could it be applied to solve this limit?
4. How does continuity of functions impact the evaluation of limits?
5. How are limits with trigonometric functions solved using similar techniques?

**Tip:** When facing limits involving square roots, multiplying by the conjugate can often help simplify the expression.

Discover how to evaluate the limit lim (h → 0) [(√(49 + h) - 7) / h] using the conjugate multiplication method. This problem focuses on key mathematical concepts such as limits, radicals, and algebraic simplification, suitable for students in Grades 11-12.

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