Evaluate the following limits:
`lim_(xrarr0)(sin(x//4))/(x)`

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{\sin\left(\frac{x}{4}\right)}{x} \), we can follow these steps:

### Step 1: Rewrite the limit
We start with the limit:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{x}{4}\right)}{x} \]
To simplify this expression, we can multiply and divide by \( \frac{1}{4} \):
\[ \lim_{x \to 0} \frac{\sin\left(\frac{x}{4}\right)}{\frac{x}{4}} \cdot \frac{1}{4} \]

### Step 2: Apply the standard limit
We know from the standard limit that:
\[ \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \]
In our case, let \( u = \frac{x}{4} \). As \( x \to 0 \), \( u \to 0 \) as well. Thus, we can rewrite our limit:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{x}{4}\right)}{\frac{x}{4}} = 1 \]

### Step 3: Combine the results
Now substituting back into our limit, we have:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{x}{4}\right)}{x} = 1 \cdot \frac{1}{4} = \frac{1}{4} \]

### Final Answer
Thus, the limit evaluates to:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{x}{4}\right)}{x} = \frac{1}{4} \]

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