Questions: lim as h approaches 0 of (sqrt(5n+4)-2)/h * ((sqrt(5n+4)+2)/(sqrt(5n+4)+2))

Transcript text: Ex-13 $\lim _{h \rightarrow 0} \frac{\sqrt{5 n+4}-2}{h} \cdot\left(\frac{\sqrt{5 n+4}+2}{\sqrt{5 n+4}+2}\right)$

Solution

Step 1: Simplifying the Expression

We start with the limit expression:

\[ \lim_{h \rightarrow 0} \frac{\sqrt{5n + 4} - 2}{h} \cdot \left(\frac{\sqrt{5n + 4} + 2}{\sqrt{5n + 4} + 2}\right) \]

To simplify, we multiply the numerator by the conjugate:

\[ \frac{(\sqrt{5n + 4} - 2)(\sqrt{5n + 4} + 2)}{h(\sqrt{5n + 4} + 2)} \]

This results in:

\[ \frac{(5n + 4) - 4}{h(\sqrt{5n + 4} + 2)} = \frac{5n}{h(\sqrt{5n + 4} + 2)} \]

Step 2: Evaluating the Limit

Now we evaluate the limit as $ h $ approaches 0:

\[ \lim_{h \rightarrow 0} \frac{5n}{h(\sqrt{5n + 4} + 2)} \]

As $ h $ approaches 0, the term $ \frac{1}{h} $ approaches infinity, leading to:

\[ \text{result} = \infty \cdot \text{sign}(\sqrt{5n + 4} - 2) \]

The limit diverges depending on the sign of $ \sqrt{5n + 4} - 2 $. Therefore, the final answer is:

\[ \boxed{\text{The limit diverges based on the sign of } \sqrt{5n + 4} - 2} \]

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