`lim_(xto0)[(-2x)/(tanx)]`, where [.] denotes greatest integer function is

Options

A. -1
B. 4
C. 5
D. None of these

Answer

To solve the limit \( \lim_{x \to 0} \left[ \frac{-2x}{\tan x} \right] \), where \([.]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Analyze the limit
We start with the expression:
\[ \lim_{x \to 0} \frac{-2x}{\tan x} \]
As \( x \) approaches 0, both the numerator and denominator approach 0. This suggests that we can apply L'Hôpital's Rule or use known limits.

### Step 2: Use the known limit
We know from calculus that:
\[ \lim_{x \to 0} \frac{\tan x}{x} = 1 \]
This implies:
\[ \lim_{x \to 0} \frac{x}{\tan x} = 1 \]
Thus, we can rewrite our limit as:
\[ \lim_{x \to 0} \frac{-2x}{\tan x} = -2 \cdot \lim_{x \to 0} \frac{x}{\tan x} = -2 \cdot 1 = -2 \]

### Step 3: Apply the greatest integer function
Now we have:
\[ \lim_{x \to 0} \frac{-2x}{\tan x} = -2 \]
Next, we apply the greatest integer function:
\[ \left[ -2 \right] = -2 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{-2} \]

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