Evaluate `lim_(xto oo)e^(x)sin (d//e^(x))`.

Answer

To evaluate the limit \( \lim_{x \to \infty} e^{x} \sin\left(\frac{d}{e^{x}}\right) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the limit expression**:
We start with the limit:
\[ L = \lim_{x \to \infty} e^{x} \sin\left(\frac{d}{e^{x}}\right) \]

2. **Substitute \( \theta \)**:
Let \( \theta = \frac{d}{e^{x}} \). As \( x \to \infty \), \( e^{x} \to \infty \) which implies \( \theta \to 0 \).

3. **Rewrite the limit**:
We can express the limit in terms of \( \theta \):
\[ L = \lim_{\theta \to 0} e^{x} \sin(\theta) \]
Since \( e^{x} = \frac{d}{\theta} \), we can rewrite:
\[ L = \lim_{\theta \to 0} \frac{d}{\theta} \sin(\theta) \]

4. **Use the limit property of sine**:
We know that as \( \theta \to 0 \):
\[ \frac{\sin(\theta)}{\theta} \to 1 \]
Therefore, we can rewrite the limit:
\[ L = d \cdot \lim_{\theta \to 0} \frac{\sin(\theta)}{\theta} = d \cdot 1 = d \]

5. **Conclusion**:
Thus, the final result is:
\[ L = d \]

### Final Answer:
\[ \lim_{x \to \infty} e^{x} \sin\left(\frac{d}{e^{x}}\right) = d \]