\lim_{{x \to 0}} \frac{x \tan \theta - \theta \tan x}{x - \theta}

To evaluate the limit \(\lim_{{x \to 0}} \frac{x \tan \theta - \theta \tan x}{x - \theta}\), let's proceed by applying L'Hôpital's Rule, since the limit has the indeterminate form \( \frac{0}{0} \).

### Step 1: Verify the indeterminate form
As \(x \to 0\):
- \(\tan(x) \approx x\) and \(\tan(\theta) \approx \theta\).
- Thus, \(x \tan \theta \approx x \theta\) and \(\theta \tan x \approx \theta x\).
- Therefore, \( \frac{x \tan \theta - \theta \tan x}{x - \theta} \approx \frac{0}{0}\), confirming that this is an indeterminate form.

### Step 2: Apply L'Hôpital's Rule
L'Hôpital's Rule states that:
\[
\lim_{{x \to a}} \frac{f(x)}{g(x)} = \lim_{{x \to a}} \frac{f'(x)}{g'(x)}
\]
if the limit is in an indeterminate form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

Define:
- \( f(x) = x \tan \theta - \theta \tan x \)
- \( g(x) = x - \theta \)

Now, differentiate \(f(x)\) and \(g(x)\) with respect to \(x\):

1. Differentiate \( f(x) \) with respect to \(x\):
\[
f'(x) = \frac{d}{dx}(x \tan \theta) - \frac{d}{dx}(\theta \tan x)
\]
\[
f'(x) = \tan \theta - \theta \sec^2(x)
\]
(Since \(\frac{d}{dx}(x \tan \theta) = \tan \theta\) and \(\frac{d}{dx}(\theta \tan x) = \theta \sec^2(x)\)).

2. Differentiate \( g(x) \) with respect to \(x\):
\[
g'(x) = \frac{d}{dx}(x - \theta) = 1
\]

### Step 3: Take the limit
Now substitute into the expression for the limit:
\[
\lim_{{x \to 0}} \frac{f'(x)}{g'(x)} = \lim_{{x \to 0}} \frac{\tan \theta - \theta \sec^2(x)}{1}
\]

Since \(\sec^2(x) \to 1\) as \(x \to 0\), we have:
\[
\lim_{{x \to 0}} (\tan \theta - \theta \sec^2(x)) = \tan \theta - \theta \cdot 1 = \tan \theta - \theta
\]

Thus, the final answer is:
\[
\lim_{{x \to 0}} \frac{x \tan \theta - \theta \tan x}{x - \theta} = \tan \theta - \theta
\]

Would you like further details or have any questions?

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### 5 Related Questions:
1. What is L'Hôpital's Rule and when is it applicable?
2. How do you differentiate trigonometric functions like \(\tan(x)\)?
3. What are the common indeterminate forms in calculus?
4. How do limits behave for trigonometric functions near zero?
5. Can we use other methods to evaluate limits involving trigonometric functions?

**Tip:** Always verify if a limit expression is in an indeterminate form before applying L'Hôpital's Rule.

Learn how to evaluate the limit \( \lim_{{x \to 0}} \frac{{x \tan \theta - \theta \tan x}}{{x - \theta}} \) using L'Hôpital's Rule. This problem involves trigonometric functions and indeterminate forms, with step-by-step solutions and explanations.

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