https://file.aiquickdraw.com/mathbot/file/1726037792627l6xymuke.jpg

The problem shown in the image involves a function \( f(x) = \frac{e^x - \sqrt{\cos x}}{2x + x^2} \), and the goal is to evaluate the limit of \( f(x) \) as \( x \to 0 \). Specifically, the question asks for the asymptotic behavior of \( f(x) \) as \( x \) approaches 0, denoted as \( \lim_{x \to 0} f(x) \).

### Breakdown of the function:
1. **Numerator**: \( e^x - \sqrt{\cos x} \)
    - As \( x \to 0 \), \( e^x \to 1 \) and \( \cos x \to 1 \), so \( \sqrt{\cos x} \to 1 \). Hence, \( e^x - \sqrt{\cos x} \to 0 \).
    - Using first-order approximations: \( e^x \approx 1 + x \) and \( \cos x \approx 1 - \frac{x^2}{2} \), so \( \sqrt{\cos x} \approx 1 - \frac{x^2}{4} \). The numerator becomes approximately:
    \[
    e^x - \sqrt{\cos x} \approx (1 + x) - \left( 1 - \frac{x^2}{4} \right) = x + \frac{x^2}{4}
    \]

2. **Denominator**: \( 2x + x^2 \)
    - This is a simple quadratic expression in \( x \), which remains as it is: \( 2x + x^2 \).

### Simplifying the expression:
We can now write \( f(x) \) as:
\[
f(x) = \frac{x + \frac{x^2}{4}}{2x + x^2}
\]
Factoring out \( x \) from both the numerator and the denominator:
\[
f(x) = \frac{x \left( 1 + \frac{x}{4} \right)}{x \left( 2 + x \right)}
\]
Cancelling \( x \) from both the numerator and denominator:
\[
f(x) = \frac{1 + \frac{x}{4}}{2 + x}
\]

As \( x \to 0 \), this expression tends to:
\[
\lim_{x \to 0} f(x) = \frac{1}{2}
\]

Thus, the correct answer for the limit is \( \boxed{\frac{1}{2}} \), which corresponds to option A.

### Would you like more details or have any questions about this explanation?

Here are five related questions to deepen your understanding:
1. How do we derive first-order approximations for \( e^x \) and \( \cos x \)?
2. What is the significance of factoring in the simplification of limits?
3. Why can we cancel terms involving \( x \) when evaluating limits?
4. How would the result change if we evaluated the limit as \( x \to \infty \)?
5. Can L'Hôpital's Rule be applied here for a faster solution?

**Tip**: Always use series expansions (like Taylor expansions) when evaluating limits involving small values of \( x \), as they provide approximations that simplify the problem.

Evaluate the limit of f(x) = (e^x - sqrt(cos x)) / (2x + x^2) as x approaches 0.

Solve the limit problem of f(x) = (e^x - sqrt(cos x)) / (2x + x^2) as x approaches 0 using Taylor expansions and asymptotic analysis. Learn the step-by-step approach to simplifying the expression.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation