Lim x -> 0  x root x + cos ^2 x + 2 e^-x / x 

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.

Learn how to evaluate the limit of the function (x sqrt(x) + cos^2(x) + 2e^-x) / x as x approaches 0. This problem involves analyzing indeterminate forms and applying L'Hôpital's rule where appropriate. Suitable for college-level students and anyone interested in calculus and limits.

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