Solve (i) `lim_(xto0^(+))[(tanx)/x]`
(ii) `lim_(xto0^(-))[(tanx)/x]`
(where [.] denotes greatest integer function)

Answer

To solve the limits given in the question, we will analyze the behavior of the function \(\frac{\tan x}{x}\) as \(x\) approaches \(0\) from the positive and negative sides.

### Step-by-Step Solution

**(i) Calculate \(\lim_{x \to 0^+} \left(\frac{\tan x}{x}\right)\)**

1. **Understanding the limit**: We need to find the limit of \(\frac{\tan x}{x}\) as \(x\) approaches \(0\) from the right (positive side).

2. **Using the known limit**: We know from calculus that:
\[ \lim_{x \to 0} \frac{\tan x}{x} = 1 \]
This limit holds true as \(x\) approaches \(0\) from either side.

3. **Behavior of \(\tan x\)**: As \(x\) approaches \(0\) from the positive side, \(\tan x\) is slightly greater than \(x\) (since \(\tan x > x\) for small positive \(x\)). Therefore, we can conclude:
\[ \frac{\tan x}{x} > 1 \quad \text{(for small } x > 0\text{)} \]

4. **Conclusion for the limit**: Thus, as \(x\) approaches \(0^+\),
\[ \frac{\tan x}{x} \to 1^+ \quad \text{(slightly greater than 1)} \]

5. **Applying the greatest integer function**: The greatest integer function \([\cdot]\) of a number slightly greater than \(1\) is:
\[ \left[\lim_{x \to 0^+} \frac{\tan x}{x}\right] = [1^+] = 1 \]

**Final Result for (i)**:
\[ \lim_{x \to 0^+} \left[\frac{\tan x}{x}\right] = 1 \]

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**(ii) Calculate \(\lim_{x \to 0^-} \left(\frac{\tan x}{x}\right)\)**

1. **Understanding the limit**: We need to find the limit of \(\frac{\tan x}{x}\) as \(x\) approaches \(0\) from the left (negative side).

2. **Using the known limit**: Again, we use the fact that:
\[ \lim_{x \to 0} \frac{\tan x}{x} = 1 \]
This limit is valid for \(x\) approaching \(0\) from both sides.

3. **Behavior of \(\tan x\)**: As \(x\) approaches \(0\) from the negative side, \(\tan x\) is also slightly greater than \(x\) (since \(\tan x > x\) for small negative \(x\)). Thus, we conclude:
\[ \frac{\tan x}{x} > 1 \quad \text{(for small } x < 0\text{)} \]

4. **Conclusion for the limit**: Therefore, as \(x\) approaches \(0^-\),
\[ \frac{\tan x}{x} \to 1^+ \quad \text{(slightly greater than 1)} \]

5. **Applying the greatest integer function**: The greatest integer function \([\cdot]\) of a number slightly greater than \(1\) is:
\[ \left[\lim_{x \to 0^-} \frac{\tan x}{x}\right] = [1^+] = 1 \]

**Final Result for (ii)**:
\[ \lim_{x \to 0^-} \left[\frac{\tan x}{x}\right] = 1 \]

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### Summary of Results
Both limits yield the same result:
\[ \lim_{x \to 0^+} \left[\frac{\tan x}{x}\right] = 1 \quad \text{and} \quad \lim_{x \to 0^-} \left[\frac{\tan x}{x}\right] = 1 \]