Evaluate the following :
`lim_(theta to 0)(sin4theta)/(tan3theta)`

Answer

To evaluate the limit \( \lim_{\theta \to 0} \frac{\sin 4\theta}{\tan 3\theta} \), we can follow these steps:

### Step 1: Identify the form of the limit
As \( \theta \) approaches 0, both the numerator and denominator approach 0:
\[ \sin 4\theta \to 0 \quad \text{and} \quad \tan 3\theta \to 0 \]
This gives us a \( \frac{0}{0} \) indeterminate form.

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that if \( \lim_{x \to c} \frac{f(x)}{g(x)} \) is of the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then:
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]
We will differentiate the numerator and the denominator with respect to \( \theta \).

### Step 3: Differentiate the numerator and denominator
1. Differentiate the numerator:
\[ \frac{d}{d\theta}(\sin 4\theta) = 4\cos 4\theta \]
2. Differentiate the denominator:
\[ \frac{d}{d\theta}(\tan 3\theta) = 3\sec^2 3\theta \]

### Step 4: Rewrite the limit using the derivatives
Now we can rewrite our limit:
\[ \lim_{\theta \to 0} \frac{\sin 4\theta}{\tan 3\theta} = \lim_{\theta \to 0} \frac{4\cos 4\theta}{3\sec^2 3\theta} \]

### Step 5: Substitute \( \theta = 0 \)
Now we can substitute \( \theta = 0 \):
\[ = \frac{4\cos(0)}{3\sec^2(0)} = \frac{4 \cdot 1}{3 \cdot 1} = \frac{4}{3} \]

### Final Answer
Thus, the limit is:
\[ \lim_{\theta \to 0} \frac{\sin 4\theta}{\tan 3\theta} = \frac{4}{3} \]

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