To solve the limit

$\lim_{{x \to 0}} \frac{{\sin(x)}}{{2x}},$

let’s break it down step by step:

1. Recognize the limit of the basic sine function: You might already know the standard limit result:

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

This fact will help simplify the problem.

2. Rearrange the expression: Try rewriting the given limit to make use of this standard limit. You can manipulate the expression by splitting it:

$\frac{{\sin(x)}}{{2x}} = \frac{1}{2} \cdot \frac{{\sin(x)}}{{x}}.$

3. Apply the limit to each part: Now, think about the limit of the product. Since the limit of $\frac{{\sin(x)}}{{x}}$ as $x \to 0$ is 1, you can compute the limit of the whole expression.

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Do you want to dive deeper into any of these steps or explore the justification behind the limit of $\frac{{\sin(x)}}{{x}}$?