Evaluate the following limits :
`Lim_(x to 0) (tan ax )/(tan bx )`

Answer

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

### Step 1: Recognize the Standard Limit
We know from standard calculus that:
\[ \lim_{x \to 0} \frac{\tan(x)}{x} = 1 \]
This means as \( x \) approaches 0, \( \tan(x) \) behaves like \( x \).

### Step 2: Rewrite the Limit
We can rewrite the limit in terms of \( ax \) and \( bx \):
\[ \lim_{x \to 0} \frac{\tan(ax)}{\tan(bx)} = \lim_{x \to 0} \frac{\tan(ax)}{ax} \cdot \frac{ax}{bx} \cdot \frac{bx}{\tan(bx)} \]

### Step 3: Apply the Standard Limit
Now we can separate the limit:
\[ \lim_{x \to 0} \frac{\tan(ax)}{ax} \cdot \lim_{x \to 0} \frac{ax}{bx} \cdot \lim_{x \to 0} \frac{bx}{\tan(bx)} \]
Using the standard limit:
\[ \lim_{x \to 0} \frac{\tan(ax)}{ax} = 1 \quad \text{and} \quad \lim_{x \to 0} \frac{bx}{\tan(bx)} = 1 \]

### Step 4: Simplify the Remaining Terms
Now, we simplify the remaining term:
\[ \lim_{x \to 0} \frac{ax}{bx} = \frac{a}{b} \]

### Step 5: Combine the Results
Putting it all together, we have:
\[ \lim_{x \to 0} \frac{\tan(ax)}{\tan(bx)} = 1 \cdot \frac{a}{b} \cdot 1 = \frac{a}{b} \]

### Final Answer
Thus, the limit evaluates to:
\[ \lim_{x \to 0} \frac{\tan(ax)}{\tan(bx)} = \frac{a}{b} \]

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