https://file.aiquickdraw.com/mathbot/file/1729348832116ja2klrj9.jpg

The limit you provided is:

\[
\lim_{{x \to 0}} \frac{e^x + e^{-x} - 2}{1 - \cos(2x)}
\]

### Step 1: Simplify the numerator
The numerator is \( e^x + e^{-x} - 2 \). This expression can be recognized as part of the expansion for the hyperbolic cosine function:

\[
e^x + e^{-x} = 2 \cosh(x)
\]

So, the numerator becomes:

\[
e^x + e^{-x} - 2 = 2 \cosh(x) - 2 = 2 (\cosh(x) - 1)
\]

Using the approximation \( \cosh(x) \approx 1 + \frac{x^2}{2} \) for small \(x\), we get:

\[
2 (\cosh(x) - 1) \approx 2 \left( \frac{x^2}{2} \right) = x^2
\]

Thus, near \( x = 0 \), the numerator approximates to \( x^2 \).

### Step 2: Simplify the denominator
The denominator is \( 1 - \cos(2x) \). Using the small angle approximation \( \cos(2x) \approx 1 - \frac{(2x)^2}{2} = 1 - 2x^2 \), we have:

\[
1 - \cos(2x) \approx 2x^2
\]

### Step 3: Compute the limit
Now, substitute these approximations back into the original limit:

\[
\lim_{{x \to 0}} \frac{x^2}{2x^2}
\]

This simplifies to:

\[
\lim_{{x \to 0}} \frac{1}{2} = \frac{1}{2}
\]

### Final Answer:
The value of the limit is \( \frac{1}{2} \).

---

Would you like further details on any of the steps or concepts? Here are a few follow-up questions to deepen your understanding:

1. What is the small-angle approximation for trigonometric functions like \( \cos(x) \)?
2. How is the hyperbolic cosine function \( \cosh(x) \) related to exponential functions?
3. Why does the numerator \( e^x + e^{-x} - 2 \) simplify using the hyperbolic cosine approximation?
4. How can L'Hopital's Rule be applied to limits of the indeterminate form \( \frac{0}{0} \)?
5. What are some common techniques for solving limits involving trigonometric functions?

**Tip**: In limits involving trigonometric and exponential functions, small-angle approximations can significantly simplify the problem near \(x = 0\).

Solve the limit lim (x→0) (e^x + e^(-x) - 2) / (1 - cos(2x))

Learn how to solve the limit lim (x→0) (e^x + e^(-x) - 2) / (1 - cos(2x)) step-by-step. This calculus problem involves the use of exponential and trigonometric functions, hyperbolic cosine expansion, and small-angle approximation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation