Question:

Step 1: Understand the condition for the limit to exist. For $\displaystyle\lim_{x \to N} f(x)$ to exist, the left and right limits must be equal. Since $f$ is defined differently on rationals and irrationals, we need: $$\lim_{x \to N, x \in \mathbb{Q}} f(x) = \lim_{x \to N, x \notin \mathbb{Q}} f(x)$$ Step 2: Evaluate the limit along rational numbers. As $x$ approaches $N$ through rational values: $$\lim_{x \to N, x \in \mathbb{Q}} f(x) = \lim_{x \to N, x \in \mathbb{Q}} \sin(\pi x) = \sin(\pi N)$$ Since $N$ is a positive integer, $\sin(\pi N) = 0$. Step 3: Evaluate the limit along irrational numbers. As $x$ approaches $N$ through irrational values (where $N > 0$): $$\lim_{x \to N, x \notin \mathbb{Q}} f(x) = \lim_{x \to N, x \notin \mathbb{Q}} \tan(\pi\sqrt{|x|}) = \tan(\pi\sqrt{N})$$ Step 4: Set up the equation for limit existence. For the limit to exist, both one-sided limits must be equal: $$\tan(\pi\sqrt{N}) = \sin(\pi N) = 0$$ Step 5: Solve for $N$. For $\tan(\pi\sqrt{N}) = 0$, we need: $$\pi\sqrt{N} = m\pi$$ where $m$ is a non-negative integer. Dividing by $\pi$: $$\sqrt{N} = m$$ Squaring both sides: $$N = m^2$$ Therefore, $N$ must be a perfect square. Step 6: List all positive perfect squares less than 100. The positive perfect squares less than 100 are: $$1^2 = 1, \quad 2^2 = 4, \quad 3^2 = 9, \quad 4^2 = 16, \quad 5^2 = 25$$ $$6^2 = 36, \quad 7^2 = 49, \quad 8^2 = 64, \quad 9^2 = 81$$ Step 7: Calculate the sum. $$\text{Sum} = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81$$ Computing step by step: $$= (1 + 9) + (4 + 16) + (25 + 36) + (49 + 64) + 81$$ $$= 10 + 20 + 61 + 113 + 81$$ $$= 285$$ The sum of all positive integers $N < 100$ for which $\displaystyle\lim_{x \to N} f(x)$ exists is $\boxed{285}$. **The answer is Option 4.**