2.4.2 Vertical Asymptotes and Discontinuities¶

Identifying vertical asymptotes from denominators and distinguishing between removable discontinuities (holes) and non-removable discontinuities.

定义¶

A vertical asymptote is a vertical line \(x = a\) where a rational function approaches infinity (positive or negative) as \(x\) approaches \(a\) from either side. Formally, \(x = a\) is a vertical asymptote of \(f(x)\) if \(\lim_{x \to a^+} f(x) = \pm\infty\) or \(\lim_{x \to a^-} f(x) = \pm\infty\). \nA discontinuity occurs at a point where a function is not continuous. There are two main types: 1. Removable Discontinuity (Hole): Occurs when both the numerator and denominator of a rational function share a common factor. The function is undefined at that point, but the limit exists and is finite. If \(f(x) = \frac{(x-a)g(x)}{(x-a)h(x)}\) where \(g(a) \neq 0\) and \(h(a) \neq 0\), then \(x = a\) is a removable discontinuity.

Non-Removable Discontinuity (Vertical Asymptote): Occurs when the denominator equals zero but the numerator does not (after canceling common factors). The function approaches infinity as \(x\) approaches this value, creating a vertical asymptote.

核心公式¶

\(x = a \text{ is a vertical asymptote if } \lim_{x \to a^+} f(x) = \pm\infty \text{ or } \lim_{x \to a^-} f(x) = \pm\infty\)
\(f(x) = \frac{(x-a)g(x)}{(x-a)h(x)} \Rightarrow x = a \text{ is a removable discontinuity with hole at } (a, \lim_{x \to a} f(x))\)
\(f(x) = \frac{p(x)}{q(x)} \text{ has vertical asymptotes where } q(x) = 0 \text{ and } p(x) \neq 0\)
\(\lim_{x \to a} f(x) = L \text{ (finite)} \Rightarrow x = a \text{ is a removable discontinuity}\)
\(\lim_{x \to a} f(x) = \pm\infty \Rightarrow x = a \text{ is a non-removable discontinuity (vertical asymptote)}\)

易错点¶

⚠️ Confusing removable and non-removable discontinuities: Students often forget to factor and cancel common factors before determining the type of discontinuity. A hole exists only when the same factor appears in both numerator and denominator; a vertical asymptote exists when the denominator is zero but the numerator is not (after simplification).
⚠️ Failing to simplify rational functions before identifying asymptotes: For example, in \(f(x) = \frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{x-1}\), students may incorrectly identify \(x = 1\) as a vertical asymptote when it is actually a removable discontinuity (hole at \((1, 2)\)).
⚠️ Ignoring the behavior of the function near the asymptote: Students may identify a vertical asymptote but fail to determine whether the function approaches \(+\infty\) or \(-\infty\) from each side, which is important for sketching accurate graphs.
⚠️ Assuming every zero of the denominator creates a vertical asymptote: Students forget that if a factor in the denominator cancels with a factor in the numerator, that point is a hole, not an asymptote. The multiplicity of factors matters—if the denominator has a higher multiplicity of a factor than the numerator, then a vertical asymptote exists.