The question you are asking to solve is:

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

This type of limit appears to be in an indeterminate form $\frac{0}{0}$, which is a typical scenario for applying L'Hopital's Rule or expanding using Taylor series. However, another simpler approach could be to use the product rule of limits along with the derivative forms of trigonometric functions.

Let's solve this step-by-step:

Step 1: Verify Indeterminate Form

As $x \to \theta$, both the numerator and denominator tend to 0, so it is indeed an indeterminate form $\frac{0}{0}$.

Step 2: Apply L'Hopital's Rule

To resolve the indeterminate form, we differentiate the numerator and denominator separately.

• Derivative of the numerator $x \tan \theta - \theta \tan x$:

  • For the first term $x \tan \theta$, differentiate with respect to $x$: $\frac{d}{dx} (x \tan \theta) = \tan \theta$.
  • For the second term $-\theta \tan x$, differentiate with respect to $x$: $\frac{d}{dx} (-\theta \tan x) = -\theta \sec^2 x$.

  Therefore, the derivative of the numerator is: $\tan \theta - \theta \sec^2 x$

• Derivative of the denominator $x - \theta$ with respect to $x$ is simply: $1$

Step 3: Evaluate the Limit

Now substitute $x = \theta$ into the derivatives:

$\lim_{x \to \theta} \frac{\tan \theta - \theta \sec^2 x}{1}$

At $x = \theta$, this becomes:

$\tan \theta - \theta \sec^2 \theta$

Thus, the limit is:

$\tan \theta - \theta \sec^2 \theta$

Final Answer:

$\boxed{\tan \theta - \theta \sec^2 \theta}$

Let me know if you'd like more details or have any questions! Here are five related questions that build on this concept:

1. How would you apply L'Hopital's rule if the limit was in the form $\infty / \infty$?
2. What happens if we change $\theta$ to a different variable in the limit process?
3. How can Taylor series expansions help in solving similar limits involving trigonometric functions?
4. What other trigonometric identities could simplify limits like this one?
5. How does L'Hopital’s rule apply when both the numerator and denominator tend to infinity?

Tip: Always check for indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before deciding on which rule or method to apply for limits.