We are given the limit:

$\lim_{{x \to 0}} \frac{1 - \cos{x}}{x}$

To evaluate this limit, we can use standard limit techniques. One common approach for this type of problem is applying a known trigonometric limit formula or L'Hopital's Rule, since direct substitution gives an indeterminate form $\frac{0}{0}$.

Method 1: Using Trigonometric Identity

First, we use the trigonometric identity:

$1 - \cos{x} = 2\sin^2\left(\frac{x}{2}\right)$

Thus, the given limit becomes:

$\lim_{{x \to 0}} \frac{2\sin^2\left(\frac{x}{2}\right)}{x}$

We now divide both the numerator and denominator by $x$:

$= \lim_{{x \to 0}} \frac{2\sin^2\left(\frac{x}{2}\right)}{x} = 2 \lim_{{x \to 0}} \frac{\sin^2\left(\frac{x}{2}\right)}{\left(\frac{x}{2}\right)} \cdot \frac{1}{2}$

Now using the standard limit:

\lim_{{x \to 0}} \frac{\sin{y}}{y} = 1 \quad \text{(where y = \frac{x}{2})},

we get:

$2 \cdot \left( \frac{1}{2} \right) = 0.$

Therefore, the value of the limit is 0.

Method 2: Using L'Hopital's Rule

Since the given limit initially results in the indeterminate form $\frac{0}{0}$, we can apply L'Hopital's Rule, which involves taking the derivative of the numerator and the denominator.

1. Derivative of the numerator $1 - \cos{x}$ is $\sin{x}$.
2. Derivative of the denominator $x$ is $1$.

Thus, applying L'Hopital's Rule, the limit becomes:

$\lim_{{x \to 0}} \frac{\sin{x}}{1} = \sin{0} = 0.$

So, the value of the limit is 0.

Let me know if you want more details or further clarification!

Here are 5 relative questions that can expand your understanding:

1. What is L'Hopital's Rule and when should it be applied in limits?
2. How can we use Taylor series to expand functions like $\cos{x}$ and evaluate limits?
3. What are some other indeterminate forms besides $\frac{0}{0}$?
4. Why is the limit $\lim_{x \to 0} \frac{\sin{x}}{x} = 1$ considered fundamental in calculus?
5. How does the small-angle approximation $\sin{x} \approx x$ near zero help in solving limits?

Tip: Memorizing common trigonometric limits and identities can greatly speed up solving problems involving limits like this!