Let $m$ be a positive integer. If $\lim_{x \to 0} |\cos x + \sin 2x + \sin 3x|^{\cot x} = e^m$, then the value of $m$ is:
Step-by-Step Solution
Key Concept: $1^\infty$ form limit using $e^{\lim f(x)\ln g(x)}$
Step 1: Identify the indeterminate form and rewrite the limit using exponentials.
We need to evaluate $L = \lim_{x \to 0}|\cos x + \sin 2x + \sin 3x|^{\cot x}$.
As $x \to 0$, the base approaches $1$ and the exponent approaches $\infty$, giving us the indeterminate form $1^\infty$. We can rewrite this using the exponential function:
$$L = e^{\lim_{x \to 0} \cot x \cdot \ln|\cos x + \sin 2x + \sin 3x|}$$
Step 2: Find the Taylor expansion of the base near $x = 0$.
We expand each trigonometric function as $x \to 0$:
- $\cos x \approx 1$
- $\sin 2x \approx 2x$
- $\sin 3x \approx 3x$
Therefore:
$$\cos x + \sin 2x + \sin 3x \approx 1 + 2x + 3x = 1 + 5x$$
Step 3: Simplify the logarithm of the base.
Using the approximation $\ln(1 + u) \approx u$ for small $u$:
$$\ln|\cos x + \sin 2x + \sin 3x| = \ln(1 + 5x) \approx 5x$$
Step 4: Find the approximation of $\cot x$ near $x = 0$.
As $x \to 0$:
$$\cot x = \frac{\cos x}{\sin x} \approx \frac{1}{x}$$
Step 5: Evaluate the exponent in the limit.
The exponent becomes:
$$\lim_{x \to 0} \cot x \cdot \ln|\cos x + \sin 2x + \sin 3x| = \lim_{x \to 0} \frac{1}{x} \cdot 5x = \lim_{x \to 0} 5 = 5$$
Step 6: State the final answer.
Therefore:
$$L = e^5$$
Comparing with the given condition $\lim_{x \to 0} |\cos x + \sin 2x + \sin 3x|^{\cot x} = e^m$, we have:
$$m = 5$$
The answer is **Option 4: 5**
Correct Answer: 4