1. **State the problem:** Find the limit $$\lim_{x \to \pi} \frac{x + \pi \sec x}{x^2 - \pi^2}.$$\n\n2. **Recall the formula and rules:** The limit is of the form $$\frac{f(x)}{g(x)}$$ where both numerator and denominator approach 0 as $x \to \pi$, so we can apply L'Hôpital's Rule.\n\n3. **Check the form:**\nEvaluate numerator at $x=\pi$: $$\pi + \pi \sec \pi = \pi + \pi \cdot (-1) = 0.$$\nEvaluate denominator at $x=\pi$: $$\pi^2 - \pi^2 = 0.$$\nSo the limit is $$\frac{0}{0}$$ indeterminate form.\n\n4. **Apply L'Hôpital's Rule:** Differentiate numerator and denominator separately.\nNumerator derivative: $$\frac{d}{dx} \left(x + \pi \sec x\right) = 1 + \pi \sec x \tan x.$$\nDenominator derivative: $$\frac{d}{dx} \left(x^2 - \pi^2\right) = 2x.$$\n\n5. **Evaluate the new limit:**\n$$\lim_{x \to \pi} \frac{1 + \pi \sec x \tan x}{2x} = \frac{1 + \pi \sec \pi \tan \pi}{2 \pi}.$$\nRecall: $$\sec \pi = -1, \quad \tan \pi = 0.$$\nSo numerator becomes $$1 + \pi \cdot (-1) \cdot 0 = 1.$$\nDenominator is $$2 \pi.$$\n\n6. **Final answer:**\n$$\lim_{x \to \pi} \frac{x + \pi \sec x}{x^2 - \pi^2} = \frac{1}{2 \pi}.$$