Mathematical definition of a cusp versus a vertical tangent:
For a continuous curve $y = f(x)$ at $x = c$:

• Vertical Tangent: Both one-sided limits of the derivative approach the same signed infinity (both $+\infty$ or both $-\infty$). Examples: $f(x) = x^{1/3}$ has $\lim_{x \to 0} f'(x) = +\infty$.
• Cusp: The one-sided limits of the derivative approach opposite signed infinities. That is, as $x \to c^-$, $f'(x) \to +\infty$ and as $x \to c^+$, $f'(x) \to -\infty$ (or vice versa). Example: $f(x) = 1 - x^{2/3}$ or $f(x) = x^{2/3}$.
  
  Option breakdown:
• Both limits tending to $+\infty$ defines an inflection point with a vertical tangent line, not a cusp.
• Both limits tending to $-\infty$ also defines a vertical tangent, not a cusp.
• Opposite infinities ($+\infty$ and $-\infty$) precisely defines a cusp where the curve reverses direction sharply.
• Finite unequal limits define a corner, not a cusp.