Solve `lim_(xto0)["sin"(|x|)/x]`, where `e[.]` denotes greatest integer function.

Answer

To solve the limit \( \lim_{x \to 0} \left[ \frac{\sin(|x|)}{x} \right] \), where \( [.] \) denotes the greatest integer function, we will analyze the limit from both the right-hand side (RHL) and the left-hand side (LHL).

### Step 1: Analyze the Right-Hand Limit (RHL)

We first consider the limit as \( x \) approaches 0 from the right (i.e., \( x \to 0^+ \)):
\[ \lim_{x \to 0^+} \frac{\sin(|x|)}{x} = \lim_{x \to 0^+} \frac{\sin(x)}{x} \]
Since \( |x| = x \) when \( x \) is positive, we can directly substitute:
\[ \lim_{x \to 0^+} \frac{\sin(x)}{x} = 1 \]

Now, applying the greatest integer function:
\[ \left[ \lim_{x \to 0^+} \frac{\sin(x)}{x} \right] = [1] = 1 \]

### Step 2: Analyze the Left-Hand Limit (LHL)

Next, we consider the limit as \( x \) approaches 0 from the left (i.e., \( x \to 0^- \)):
\[ \lim_{x \to 0^-} \frac{\sin(|x|)}{x} = \lim_{x \to 0^-} \frac{\sin(-x)}{x} \]
Since \( |x| = -x \) when \( x \) is negative, we can rewrite this as:
\[ \lim_{x \to 0^-} \frac{-\sin(x)}{x} = -\lim_{x \to 0^-} \frac{\sin(x)}{x} = -1 \]

Now, applying the greatest integer function:
\[ \left[ \lim_{x \to 0^-} \frac{-\sin(x)}{x} \right] = [-1] = -1 \]

### Step 3: Conclusion

Since the right-hand limit (RHL) and the left-hand limit (LHL) are not equal:
\[ \lim_{x \to 0^+} \left[ \frac{\sin(|x|)}{x} \right] = 1 \quad \text{and} \quad \lim_{x \to 0^-} \left[ \frac{\sin(|x|)}{x} \right] = -1 \]
The overall limit does not exist.

### Final Answer
The limit does not exist.

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