$lim_{x\rightarrow\infty}\frac{x^{n}}{e^{x}}=0,$ (n integer), for:

Question

Accepted Answer

To understand why Option 4 is the correct answer, we need to analyze the behavior of the limit $ \lim_{x \rightarrow \infty} \frac{x^n}{e^x} = 0 $ for all integer values of $ n $.

### Explanation for Option 4:

1. **Exponential Growth vs. Polynomial Growth**: The exponential function $ e^x $ grows much faster than any polynomial function $ x^n $ as $ x $ approaches infinity. This is a fundamental property of exponential functions compared to polynomial functions.

2. **L'Hôpital's Rule**: For positive integers $ n $, we can apply L'Hôpital's Rule repeatedly (n times) to the limit $ \lim_{x \rightarrow \infty} \frac{x^n}{e^x} $. Each application of L'Hôpital's Rule reduces the power of $ x $ by 1, while the exponential function remains unchanged. Eventually, the numerator becomes a constant, and the limit evaluates to 0.

3. **Negative and Zero Powers**: For $ n = 0 $, the expression becomes $ \frac{1}{e^x} $, which clearly tends to 0 as $ x \rightarrow \infty $. For negative integers $ n $, the expression $ x^n $ represents $ \frac{1}{x^{|n|}} $, which also tends to 0 as $ x \rightarrow \infty $ because the denominator grows without bound.

4. **Conclusion**: Since the limit evaluates to 0 for all integer values of $ n $ (positive, negative, and zero), Option 4, which states "all values of n," is correct.

### Explanation for Why Other Options are Incorrect:

1. **Option 1 (no value of n)**: This option is incorrect because, as demonstrated, the limit is 0 for all integer values of $ n $. There is no integer $ n $ for which the limit does not evaluate to 0.

2. **Option 2 (only positive values of n)**: This option is incorrect because it excludes zero and negative integers. As shown, the limit is 0 for $ n = 0 $ and negative integers as well.

3. **Option 3 (only negative values of n)**: This option is incorrect because it excludes zero and positive integers. The limit is 0 for positive integers and zero, as demonstrated.

In summary, the correct answer is Option 4 because the limit $ \lim_{x \rightarrow \infty} \frac{x^n}{e^x} = 0 $ holds true for all integer values of $ n $, encompassing positive, negative, and zero values.