The value of `lim_(xto0) ([(100x)/(sinx)]+[(99sinx)/(x)])` (where `[.]` represents the greatest integral function) is

Options

A. `199`
B. `198`
C. `0`
D. none of these

Explanation

To find the value of

\[ \lim_{x \to 0} \left( \left\lfloor \frac{100x}{\sin x} \right\rfloor + \left\lfloor \frac{99 \sin x}{x} \right\rfloor \right) \]

we will analyze each term separately as \( x \) approaches 0.

### Step 1: Analyze \(\frac{100x}{\sin x}\)

Using the limit property:

\[ \lim_{x \to 0} \frac{x}{\sin x} = 1 \]

we can say:

\[ \lim_{x \to 0} \frac{100x}{\sin x} = 100 \cdot \lim_{x \to 0} \frac{x}{\sin x} = 100 \cdot 1 = 100 \]

As \( x \) approaches 0, \(\frac{100x}{\sin x}\) approaches 100 from above (since \( \sin x < x \) for small \( x \)), which means:

\[ \frac{100x}{\sin x} \approx 100 + \epsilon \quad (\text{where } \epsilon \text{ is a small positive number}) \]

### Step 2: Apply the Greatest Integer Function

Since \(\frac{100x}{\sin x}\) approaches 100 from above, we have:

\[ \left\lfloor \frac{100x}{\sin x} \right\rfloor = 100 \]

for sufficiently small \( x \).

### Step 3: Analyze \(\frac{99 \sin x}{x}\)

Using the limit property:

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]

we can say:

\[ \lim_{x \to 0} \frac{99 \sin x}{x} = 99 \cdot \lim_{x \to 0} \frac{\sin x}{x} = 99 \cdot 1 = 99 \]

As \( x \) approaches 0, \(\frac{99 \sin x}{x}\) approaches 99 from below (since \( \sin x < x \)), which means:

\[ \frac{99 \sin x}{x} \approx 99 - \delta \quad (\text{where } \delta \text{ is a small positive number}) \]

### Step 4: Apply the Greatest Integer Function

Since \(\frac{99 \sin x}{x}\) approaches 99 from below, we have:

\[ \left\lfloor \frac{99 \sin x}{x} \right\rfloor = 98 \]

for sufficiently small \( x \).

### Step 5: Combine the Results

Now, we combine the results from Steps 2 and 4:

\[ \lim_{x \to 0} \left( \left\lfloor \frac{100x}{\sin x} \right\rfloor + \left\lfloor \frac{99 \sin x}{x} \right\rfloor \right) = 100 + 98 = 198 \]

### Final Answer

Thus, the value of the limit is

\[ \boxed{198} \]