Evaluate the following limits:
`lim_(xrarr0)(tan3x)/(tan5x)`

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{\tan 3x}{\tan 5x} \), we can follow these steps:

### Step 1: Rewrite the limit
We start by rewriting the limit in a form that allows us to apply the known limit property of the tangent function. We know that:

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

for any constant \( k \). Therefore, we can manipulate our limit as follows:

\[ \lim_{x \to 0} \frac{\tan 3x}{\tan 5x} = \lim_{x \to 0} \frac{\tan 3x}{3x} \cdot \frac{3x}{5x} \cdot \frac{5x}{\tan 5x} \]

### Step 2: Break down the limit
Now we can break this limit into three parts:

\[ \lim_{x \to 0} \frac{\tan 3x}{3x} \cdot \lim_{x \to 0} \frac{3x}{5x} \cdot \lim_{x \to 0} \frac{5x}{\tan 5x} \]

### Step 3: Evaluate each limit
1. **First limit**: \( \lim_{x \to 0} \frac{\tan 3x}{3x} = 1 \)
2. **Second limit**: \( \lim_{x \to 0} \frac{3x}{5x} = \frac{3}{5} \)
3. **Third limit**: \( \lim_{x \to 0} \frac{5x}{\tan 5x} = 1 \)

### Step 4: Combine the results
Now we can combine these results:

\[ \lim_{x \to 0} \frac{\tan 3x}{\tan 5x} = 1 \cdot \frac{3}{5} \cdot 1 = \frac{3}{5} \]

### Final Answer
Thus, the limit evaluates to:

\[ \lim_{x \to 0} \frac{\tan 3x}{\tan 5x} = \frac{3}{5} \]

---