https://file.aiquickdraw.com/mathbot/file/17266188406340yg5w0wz.jpg

The given image contains the following mathematical expression:

\[
\lim_{x \to 0} \frac{1 - \cos(x)}{x}
\]

To evaluate this limit, we can apply L'Hopital's Rule, since directly substituting \(x = 0\) results in an indeterminate form \(\frac{0}{0}\).

### Step-by-Step Solution:

1. **Apply L'Hopital's Rule**: This rule is used for limits of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). It states that:

\[
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}
\]

if the limit on the right exists.

In this case, both the numerator \(1 - \cos(x)\) and the denominator \(x\) approach 0 as \(x \to 0\). So we can apply L'Hopital's Rule by differentiating the numerator and the denominator:

- Derivative of the numerator: \(\frac{d}{dx}[1 - \cos(x)] = \sin(x)\)
- Derivative of the denominator: \(\frac{d}{dx}[x] = 1\)

2. **Rewrite the limit**:

\[
\lim_{x \to 0} \frac{1 - \cos(x)}{x} = \lim_{x \to 0} \frac{\sin(x)}{1}
\]

3. **Evaluate the limit**:

As \(x \to 0\), \(\sin(x) \to 0\). Hence:

\[
\lim_{x \to 0} \sin(x) = 0
\]

So, the value of the original limit is:

\[
\boxed{0}
\]

Would you like more details, or do you have any specific questions?

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Here are five questions to extend this topic:

1. What is L'Hopital's Rule, and when can it be applied?
2. How would the limit change if the denominator were \(x^2\) instead of \(x\)?
3. Can you explain why the function \(\frac{1 - \cos(x)}{x}\) approaches 0 as \(x \to 0\)?
4. How can you approximate \(\cos(x)\) for small values of \(x\) using a Taylor series?
5. What are some alternative methods to evaluate trigonometric limits without L'Hopital's Rule?

**Tip:** Always check if the limit leads to an indeterminate form like \(\frac{0}{0}\) before applying L'Hopital's Rule!

Evaluate the limit lim (x -> 0) (1 - cos(x)) / x.

Learn how to evaluate the limit of (1 - cos(x)) / x as x approaches 0 using L'Hopital's Rule. This problem covers key calculus concepts, including limits, trigonometric functions, and derivative rules.

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