Mathematics

A vertical asymptote at $x=a$ occurs when $f(x)$ grows without bound as you approach $a$ from either side—formally, when $\lim_{x \to a^+} f(x) = \pm\infty$ or $\lim_{x \to a^-} f(x) = \pm\infty$. The key insight is that the function doesn't need to be defined at $x=a$ itself; what matters is that the output explodes to infinity, not that the function reaches a finite value. This infinite limit is the defining signature that separates a vertical asymptote from a removable hole or other discontinuity.

A vertical asymptote at $x=a$ occurs when $f(x)$ grows without bound as you approach $a$ from either side—formally, when $\lim_{x \to a^+} f(x) = \pm\infty$ or $\lim_{x \to a^-} f(x) = \pm\infty$. The key insight is that the function doesn't need to be defined at $x=a$ itself; what matters is that the output explodes to infinity, not that the function reaches a finite value. This infinite limit is the defining signature that separates a vertical asymptote from a removable hole or other discontinuity.