1. **State the problem:** Find the limit $$\lim_{t \to 0} \tan t$$ using L'Hospital's rule. 2. **Recall the formula and rule:** L'Hospital's rule applies to limits of the form $$\frac{0}{0}$$ or $$\frac{\infty}{\infty}$$. It states that $$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$ if the latter limit exists. 3. **Rewrite the limit:** We can write $$\tan t = \frac{\sin t}{\cos t}$$, but this is not a quotient tending to $$\frac{0}{0}$$ or $$\frac{\infty}{\infty}$$ as $$t \to 0$$. Instead, consider the limit directly. 4. **Evaluate directly:** Since $$\tan 0 = 0$$, the limit is simply $$0$$. 5. **Alternatively, use the known limit:** $$\lim_{t \to 0} \frac{\sin t}{t} = 1$$ and $$\lim_{t \to 0} \cos t = 1$$, so $$\lim_{t \to 0} \tan t = \lim_{t \to 0} \frac{\sin t}{\cos t} = 0$$. **Final answer:** $$\boxed{0}$$