To simplify the expression:

$\frac{1 - \frac{2}{t}}{\frac{e^{\sqrt{t}}}{2\sqrt{t}}}$

we can rewrite it as follows:

1. Start by simplifying the fraction: $= \left(1 - \frac{2}{t}\right) \cdot \frac{2\sqrt{t}}{e^{\sqrt{t}}}$

2. Distribute $2\sqrt{t}$ in the numerator: $= \frac{2\sqrt{t} - \frac{4\sqrt{t}}{t}}{e^{\sqrt{t}}}$

3. Simplify $\frac{4\sqrt{t}}{t}$ as $\frac{4}{\sqrt{t}}$: $= \frac{2\sqrt{t} - \frac{4}{\sqrt{t}}}{e^{\sqrt{t}}}$

4. Combine terms in the numerator under a common denominator: $= \frac{\frac{2t - 4}{\sqrt{t}}}{e^{\sqrt{t}}}$

5. Rewrite it as a single fraction: $= \frac{2t - 4}{\sqrt{t} \cdot e^{\sqrt{t}}}$

The simplified form is:

$\frac{2t - 4}{\sqrt{t} \cdot e^{\sqrt{t}}}$

Would you like a further breakdown of each step?

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Here are five related questions to expand on this topic:

1. How would this expression change if $t \to \infty$?
2. What if we wanted to find the derivative of this expression with respect to $t$?
3. How can we use L'Hôpital's Rule if we encounter an indeterminate form in similar expressions?
4. How does the presence of $e^{\sqrt{t}}$ affect the limit of this expression as $t \to \infty$?
5. How could this expression be interpreted in the context of calculus or physics?

Tip: Always look for common factors to simplify complex fractions, especially when working with exponentials and square roots.