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To solve the limit

\[
\lim_{{t \to 0}} \frac{e^{2t} - 1}{\sin t},
\]

let's go through it step by step.

### Step 1: Determine if L'Hôpital's Rule is Necessary

As \( t \to 0 \):
1. The numerator \( e^{2t} - 1 \) approaches \( 0 \) (since \( e^0 = 1 \)).
2. The denominator \( \sin t \) also approaches \( 0 \) (since \( \sin 0 = 0 \)).

This gives us a \( \frac{0}{0} \) indeterminate form, so we can apply L'Hôpital's Rule, which involves differentiating the numerator and the denominator.

### Step 2: Apply L'Hôpital's Rule

Differentiate the numerator and denominator with respect to \( t \):
- The derivative of \( e^{2t} - 1 \) is \( 2e^{2t} \).
- The derivative of \( \sin t \) is \( \cos t \).

Thus, the limit becomes:

\[
\lim_{{t \to 0}} \frac{2e^{2t}}{\cos t}.
\]

### Step 3: Evaluate the Limit

Now, substitute \( t = 0 \) directly:

\[
\frac{2e^{2 \cdot 0}}{\cos 0} = \frac{2 \cdot 1}{1} = 2.
\]

### Conclusion

The value of the limit is:

\[
\boxed{2}.
\]

Would you like a more detailed explanation of each step or have any questions?

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Here are 5 additional questions to expand on the concepts involved:

1. How do we determine when to apply L'Hôpital's Rule?
2. What other techniques could we use if L'Hôpital's Rule did not apply here?
3. Why is the indeterminate form necessary for using L'Hôpital’s Rule?
4. How does the exponential function behave as \( t \to 0 \) in this context?
5. What are some common pitfalls when using trigonometric functions in limits?

**Tip:** Always check if substituting values directly could work before applying more complex methods like L'Hôpital’s Rule.

Find the limit: Use L'Hôpital's Rule where appropriate. If there is a more elementary method, consider using it. If L'Hôpital's Rule doesn't apply, explain why.

\[\lim_{{t \to 0}} \frac{e^{2t} - 1}{\sin t}\]

Explore a step-by-step solution to the limit \( \lim_{{t \to 0}} \frac{e^{2t} - 1}{\sin t}\) using L'Hôpital's Rule. This problem covers essential calculus concepts, including exponential and trigonometric functions. Ideal for college-level calculus students.

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