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1.3 Estimating Limit Values from Graphs

Keywords

One-sided Limit 单侧极限 • Limit Does Not Exist (DNE) 极限不存在

Jump Discontinuity 跳跃间断 • Unbounded 无界 / 垂直渐近线 • Oscillation 震荡

Issues of Scale 尺度/缩放问题

One-Sided Limits

In section 1.2, we established that a limit is about approaching a value. But what if a function approaches different values depending on which direction you come from? This introduces the concept of one-sided limits.

  • Left-hand limit: The value \(f(x)\) approaches as \(x\) approaches \(c\) from values less than \(c\).
\[ \lim_{x \to c^-} f(x) = L \]
  • Right-hand limit: The value \(f(x)\) approaches as \(x\) approaches \(c\) from values greater than \(c\).
\[ \lim_{x \to c^+} f(x) = R \]

AP Essential Knowledge / LIM-1.C.1 & LIM-1.C.2

The general limit \(\lim\limits_{x \to c} f(x)\) exists if and only if both the left-hand and right-hand limits exist and are equal to the same real number.

That is, \(\lim\limits_{x \to c^-} f(x) = \lim\limits_{x \to c^+} f(x) = L\).

Estimating Limits from a Graph

Let's look at the piecewise function below.

  • As \(x \to 2^-\), the graph approaches a \(y\)-value of \(1\). So, \(\lim\limits_{x \to 2^-} f(x) = 1\).
  • As \(x \to 2^+\), the graph approaches a \(y\)-value of \(3\). So, \(\lim\limits_{x \to 2^+} f(x) = 3\).
  • Because \(1 \neq 3\), the general limit Does Not Exist (DNE).

When Limits Fail to Exist (DNE)

According to LIM-1.C.4, there are three primary graphical behaviors where a limit fails to exist at a particular \(x\)-value. On the AP Exam, you must be able to visually identify these three scenarios:

1. Left Limit ≠ Right Limit (Jump Behavior)

As we saw in the previous example, if the two sides don't meet at the same \(y\)-value, the limit does not exist.

2. Unbounded Behavior (Vertical Asymptotes)

Consider the graph of \(f(x) = \frac{1}{(x-1)^2}\) as \(x\) approaches \(1\).

As \(x \to 1\), the function values grow infinitely large. Since infinity is not a specific "real number", the limit Does Not Exist (DNE). (Note: Sometimes we write \(\lim\limits_{x \to 1} f(x) = \infty\) to describe the behavior, but formally, the limit still does not exist).

3. Oscillating Behavior

Consider the bizarre function \(f(x) = \sin(\frac{1}{x})\) as \(x\) approaches \(0\).

As \(x\) gets closer to \(0\), the function oscillates infinitely fast between \(-1\) and \(1\). It never settles on a single value, so the limit DNE.

The Danger of Graphing Technology

AP Essential Knowledge / LIM-1.C.3

Because of issues of scale, graphical representations of functions may miss important function behavior.

Never trust your calculator blindly! A graphing calculator draws graphs by connecting discrete pixels. If you graph \(f(x) = \frac{x^2 - 4}{x - 2}\) on a standard window, it looks exactly like the solid line \(y = x + 2\). The calculator's resolution is not fine enough to show the infinitely small hole (removable discontinuity) at \(x = 2\).

You must use analytical methods (which we will learn in Section 1.5) to confirm what the graph is implying, or zoom in extremely close to see if the technology is hiding a hole or intense oscillation.

Quiz

Test your visual limit-reading skills!