1.14 Connecting Infinite Limits and Vertical Asymptotes

Keywords

Infinite Limit 无穷极限 • Vertical Asymptote 垂直渐近线

Unbounded Behavior 无界行为 • Asymptotic Behavior 渐近行为

Infinite Limits

An infinite limit occurs when the values of a function increase or decrease without bound as \(x\) approaches a specific value \(c\).

核心要点：Infinite Limits (LIM-2.D.1)

We write \(\lim\limits_{x \to c} f(x) = \infty\) or \(\lim\limits_{x \to c} f(x) = -\infty\) to describe the behavior of the function.

Note: Technically, if a limit equals \(\infty\), the limit still Does Not Exist (DNE) because infinity is not a real number. However, using the \(\infty\) symbol provides more specific information about the function's behavior than just saying "DNE."

Defining Vertical Asymptotes

A vertical asymptote is a vertical line that the graph of a function approaches but never touches as \(x\) approaches a certain value.

核心要点：Vertical Asymptote Definition (LIM-2.D.2)

The line \(x = c\) is a vertical asymptote of the graph of \(f\) if the limit from either the left or the right is infinite. That is:

\[ \lim_{x \to c^+} f(x) = \pm\infty \quad \text{or} \quad \lim_{x \to c^-} f(x) = \pm\infty \]

Example: \(f(x) = \frac{1}{x-2}\)

As \(x\) approaches \(2\) from the right, the values of \(f(x)\) grow larger and larger. As \(x\) approaches \(2\) from the left, they become increasingly negative.

\[ \lim_{x \to 2^+} \frac{1}{x-2} = \infty, \quad \lim_{x \to 2^-} \frac{1}{x-2} = -\infty \]

Because at least one of the one-sided limits is infinite, the line \(x=2\) is a vertical asymptote.

Finding Asymptotes Analytically

For rational functions, vertical asymptotes typically occur at \(x\)-values that make the denominator zero but the numerator non-zero.

\(\frac{0}{0}\) (Indeterminate): Usually indicates a hole (removable discontinuity).
\(\frac{k}{0}\) (where \(k \neq 0\)): Indicates an infinite limit and therefore a vertical asymptote.

Extension: One-Sided vs. Two-Sided Infinite Limits

Consider \(f(x) = \frac{1}{(x-2)^2}\). In this case, both the left-hand and right-hand limits approach \(+\infty\).

\[ \lim_{x \to 2^-} \frac{1}{(x-2)^2} = \infty \quad \text{and} \quad \lim_{x \to 2^+} \frac{1}{(x-2)^2} = \infty \]

Therefore, we can say the general limit \(\lim\limits_{x \to 2} \frac{1}{(x-2)^2} = \infty\). Even though it's still technically DNE, the behavior is consistent from both sides.