`lim_(x to 0) (K sin x)/(lx + mx cos x)` is equal to

Options

A. `(k)/(l + m)`
B. `(l + m)/(k)`
C. 0
D. `(k)/(l)`

Answer

To solve the limit \(\lim_{x \to 0} \frac{K \sin x}{lx + mx \cos x}\), we can follow these steps:

### Step 1: Factor out \(x\) from the denominator
We start by rewriting the expression in a more manageable form. We can factor \(x\) out of the denominator:

\[ \lim_{x \to 0} \frac{K \sin x}{lx + mx \cos x} = \lim_{x \to 0} \frac{K \sin x}{x(l + m \cos x)} \]

### Step 2: Apply the limit
Now we can separate the limit into two parts:

\[ = \lim_{x \to 0} \frac{K \sin x}{x} \cdot \frac{1}{l + m \cos x} \]

### Step 3: Use the standard limit
We know from the standard limit that:

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

Thus, we can substitute this into our expression:

\[ = K \cdot 1 \cdot \frac{1}{l + m \cos(0)} \]

### Step 4: Evaluate the cosine term
Since \(\cos(0) = 1\), we can simplify further:

\[ = K \cdot \frac{1}{l + m \cdot 1} = \frac{K}{l + m} \]

### Final Answer
Therefore, the limit is:

\[ \lim_{x \to 0} \frac{K \sin x}{lx + mx \cos x} = \frac{K}{l + m} \]

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