Questions: Determine the following limit. lim as theta approaches infinity of (sin 12 theta) / (11 theta) Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. lim as theta approaches infinity of (sin 12 theta) / (11 theta) = (Simplify your answer.) B. The limit does not exist and is neither -∞ nor ∞.

Determine the following limit.

lim as theta approaches infinity of (sin 12 theta) / (11 theta)

Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. lim as theta approaches infinity of (sin 12 theta) / (11 theta) = (Simplify your answer.)

B. The limit does not exist and is neither -∞ nor ∞.

Transcript text: Determine the following limit. \[ \lim _{\theta \rightarrow \infty} \frac{\sin 12 \theta}{11 \theta} \] Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. $\lim _{\theta \rightarrow \infty} \frac{\sin 12 \theta}{11 \theta}=\square$ (Simplify your answer.) $\square$ B. The limit does not exist and is neither $-\infty$ nor $\infty$.

Solution

Solution Steps

Step 1: Define the Limit

We need to evaluate the limit: \[ \lim _{\theta \rightarrow \infty} \frac{\sin 12 \theta}{11 \theta} \]

Step 2: Analyze the Numerator and Denominator

The numerator $\sin 12 \theta$ oscillates between $-1$ and $1$ for all values of $\theta$. The denominator $11 \theta$ increases without bound as $\theta$ approaches infinity.

Step 3: Apply the Squeeze Theorem

Since $-1 \leq \sin 12 \theta \leq 1$, we can bound the fraction: \[ -\frac{1}{11 \theta} \leq \frac{\sin 12 \theta}{11 \theta} \leq \frac{1}{11 \theta} \] As $\theta$ approaches infinity, both $-\frac{1}{11 \theta}$ and $\frac{1}{11 \theta}$ approach $0$.

Step 4: Conclude the Limit

By the Squeeze Theorem, we conclude that: \[ \lim _{\theta \rightarrow \infty} \frac{\sin 12 \theta}{11 \theta} = 0 \]