Question:

Step 1: Evaluate $\displaystyle\lim_{x \to 0^+} f(x)$. For $x > 0$ near $0$, $\{x\} = x \to 0$ and $|1 - |x|| = 1 - x \to 1$. So $f(x) = \min(x, 1-x) \to \min(0, 1) = 0$. Thus $\displaystyle\lim_{x \to 0^+} f(x) = 0 \neq 1$. Option (a) is <b>incorrect</b>. Step 2: Evaluate $\displaystyle\lim_{x \to (3/4)^-} f(x)$. For $x$ slightly less than $3/4$ (and $x > 0$): $\{x\} = x - 0 = x \to 3/4$ and $|1 - |x|| = 1 - x \to 1/4$. So $f(x) = \min(x, 1-x) \to \min(3/4, 1/4) = 1/4$. Thus $\displaystyle\lim_{x \to (3/4)^-} f(x) = \dfrac{1}{4}$. Option (b) is <b>correct</b>. Step 3: Evaluate $f\!\left(\dfrac{-5}{2}\right)$. Since $x = -5/2 \leq 0$: $f(-5/2) = \max\left((-5/2)^2, 1\right) = \max(25/4, 1) = 25/4$. Step 4: Evaluate $f\!\left(f\!\left(\dfrac{-5}{2}\right)\right) = f\!\left(\dfrac{25}{4}\right)$. Since $25/4 > 0$: $\{25/4\} = \{6.25\} = 0.25 = 1/4$ and $|1 - |25/4|| = |1 - 25/4| = |{-21/4}| = 21/4$. So $f(25/4) = \min(1/4, 21/4) = 1/4$. Thus $f\!\left(f\!\left(\dfrac{-5}{2}\right)\right) = \dfrac{1}{4}$. Option (c) is <b>correct</b>. Step 5: Evaluate $f(-100)$. Since $x = -100 \leq 0$: $f(-100) = \max((-100)^2, 1) = \max(10000, 1) = 10000$. Step 6: Evaluate $f(f(-100)) = f(10000)$. Since $10000 > 0$: $\{10000\} = 0$ and $|1 - 10000| = 9999$. So $f(10000) = \min(0, 9999) = 0$. Thus $f(f(-100)) = 0$. Option (d) gives $f(f(-100)) = 0$, which is <b>correct</b>. Step 7: The correct options are (b) and (c), i.e., options 2 and 3.