To find the limit of $\frac{x \sqrt{x} + \cos^2(x) + 2e^{-x}}{x}$ as $x$ approaches 0, we need to carefully analyze the behavior of each term in the numerator.

The limit can be written as:

$\lim_{x \to 0} \frac{x \sqrt{x} + \cos^2(x) + 2e^{-x}}{x}$

Analyzing each term:

1. Term $x \sqrt{x}$: $x \sqrt{x} = x \cdot x^{1/2} = x^{3/2}$ As $x \to 0$, $x^{3/2} \to 0$.

2. Term $\cos^2(x)$: $\cos^2(x)$ As $x \to 0$, $\cos^2(0) = 1$.

3. Term $2e^{-x}$: $2e^{-x}$ As $x \to 0$, $2e^{-0} = 2$.

Combining the terms:

We have:

$\lim_{x \to 0} \frac{x^{3/2} + \cos^2(x) + 2e^{-x}}{x}$

This becomes:

$\lim_{x \to 0} \frac{x^{3/2}}{x} + \frac{\cos^2(x)}{x} + \frac{2e^{-x}}{x}$

Simplifying each term separately:

1. First term:

$\frac{x^{3/2}}{x} = x^{1/2}$ As $x \to 0$, $x^{1/2} \to 0$.

2. Second term:

$\frac{\cos^2(x)}{x}$ As $x \to 0$, $\frac{1}{x}$ approaches infinity because $\cos^2(x)$ approaches 1, making the term diverge to infinity.

3. Third term:

$\frac{2e^{-x}}{x}$ As $x \to 0$, $2e^{-x}$ approaches 2, making the term diverge to infinity.

Conclusion:

The second and third terms both diverge to infinity, implying that the overall limit does not exist in the conventional sense. Therefore, the limit:

$\lim_{x \to 0} \frac{x \sqrt{x} + \cos^2(x) + 2e^{-x}}{x}$

does not exist because the terms $\frac{\cos^2(x)}{x}$ and $\frac{2e^{-x}}{x}$ both approach infinity as $x$ approaches 0.

Do you want any more details or have any specific questions about this problem?

Here are five more questions you might be interested in:

1. How do you find the limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0?
2. What is the derivative of $e^{x^2}$?
3. How do you evaluate the limit of $\frac{\ln(x)}{x}$ as $x$ approaches infinity?
4. Can you find the Taylor series expansion for $\cos(x)$ around $x = 0$?
5. What is L'Hôpital's rule and how is it used to evaluate limits?

Tip: When evaluating limits that lead to indeterminate forms, consider using L'Hôpital's rule, which involves taking the derivative of the numerator and denominator to find the limit.