To verify this equation, we need to simplify $\frac{1 - \frac{2}{t}}{\frac{e^{\sqrt{t}}}{2\sqrt{t}}}$ and check if it equals $-\frac{2}{e}$.

Let's start by simplifying the expression on the left side step by step.

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

1. Rewrite the expression: We can rewrite it as: $\left(1 - \frac{2}{t}\right) \div \left(\frac{e^{\sqrt{t}}}{2\sqrt{t}}\right)$ which is equivalent to: $\left(1 - \frac{2}{t}\right) \cdot \frac{2\sqrt{t}}{e^{\sqrt{t}}}$

2. Distribute $\frac{2\sqrt{t}}{e^{\sqrt{t}}}$ over each term in the numerator: $\frac{2\sqrt{t}}{e^{\sqrt{t}}} \cdot 1 - \frac{2\sqrt{t}}{e^{\sqrt{t}}} \cdot \frac{2}{t}$

3. Simplify each term separately:

   • The first term is: $\frac{2\sqrt{t}}{e^{\sqrt{t}}}$
   • For the second term, simplify $\frac{2\sqrt{t} \cdot 2}{t \cdot e^{\sqrt{t}}}$: $\frac{4\sqrt{t}}{t e^{\sqrt{t}}} = \frac{4}{\sqrt{t} e^{\sqrt{t}}}$

4. Combine terms: Now, we have: $\frac{2\sqrt{t}}{e^{\sqrt{t}}} - \frac{4}{\sqrt{t} e^{\sqrt{t}}} = \frac{2\sqrt{t} - \frac{4}{\sqrt{t}}}{e^{\sqrt{t}}}$

5. Combine into a single fraction in the numerator: Rewrite $2\sqrt{t} - \frac{4}{\sqrt{t}}$ as: $\frac{2t - 4}{\sqrt{t}}$ so the expression becomes: $\frac{\frac{2t - 4}{\sqrt{t}}}{e^{\sqrt{t}}} = \frac{2t - 4}{\sqrt{t} e^{\sqrt{t}}}$

6. Factor out 2 in the numerator: $\frac{2(t - 2)}{\sqrt{t} e^{\sqrt{t}}}$

This simplification process does not directly lead to $-\frac{2}{e}$ unless there are specific values of $t$ that satisfy it. For a general $t$, the expression does not simplify to $-\frac{2}{e}$. However, if there is a specific value of $t$ intended in the problem, we can substitute it to check if the equation holds.

Would you like further clarification or to proceed with evaluating specific values of $t$?

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Here are some follow-up questions that may deepen understanding:

1. What specific value of $t$ might satisfy this equation if any?
2. How does simplifying complex expressions help in solving equations?
3. What other methods could be used to verify equivalency for complex fractions?
4. Is there a graphical approach to verify this equality visually?
5. How does the behavior of exponential and radical functions affect limits in similar expressions?

Tip: When simplifying complex expressions, isolating terms or factoring can often reveal hidden simplifications.