Exploring Types of | AP Calc AB Unit 1 Study Guide

The Core Idea: Exploring Types of Discontinuities

While continuity describes a function whose graph is a single, unbroken curve, many functions have "breaks." This topic moves beyond simply identifying if a function is discontinuous and focuses on classifying the nature of these breaks. Understanding the type of discontinuity provides deeper insight into the behavior of a function at a specific point.

By analyzing the limits from the left and right, and comparing them to the actual value of the function at the point, we can categorize any discontinuity into one of three types: removable (a single point "hole"), jump (a sudden leap from one value to another), or infinite (a vertical asymptote where the function's value grows without bound). This classification is based entirely on the specific limiting behavior of the function at the point of discontinuity.

Key Definitions

The classification of a discontinuity at a point $x = c$ is determined by the existence and relationship of the one-sided limits and the function's value $f (c)$ .

Removable Discontinuity: A function $f$ has a removable discontinuity at $x = c$ if the two-sided limit $lim_{x \to c} f (x)$ exists, but is not equal to the function's value $f (c)$ . This can happen because $f (c)$ is undefined or because $f (c)$ is defined as a different value than the limit. This type of discontinuity is called "removable" because the break could be fixed by redefining the function's value at the single point $x = c$ to be equal to the limit.
Jump Discontinuity: A function $f$ has a jump discontinuity at $x = c$ if the limit from the left, $lim_{x \to c^{-}} f (x)$ , and the limit from the right, $lim_{x \to c^{+}} f (x)$ , both exist as finite numbers, but they are not equal to each other. The graph "jumps" from one finite value to another.
Infinite Discontinuity: A function $f$ has an infinite discontinuity at $x = c$ if either the left-hand limit or the right-hand limit (or both) is infinite. That is, $lim_{x \to c^{-}} f (x) = \pm \infty$ or $lim_{x \to c^{+}} f (x) = \pm \infty$ . This type of discontinuity corresponds to a vertical asymptote at $x = c$ .

Understanding the Limit Conditions

The type of discontinuity at a point $x = c$ is a direct consequence of the three conditions required for continuity:

$f (c)$ must be defined.
$lim_{x \to c} f (x)$ must exist.
$lim_{x \to c} f (x) = f (c)$ .

We can use the failure of these conditions to classify the discontinuity:

If Condition 2 fails: The two-sided limit $lim_{x \to c} f (x)$ does not exist.
- If $lim_{x \to c^{-}} f (x)$ and $lim_{x \to c^{+}} f (x)$ both exist as different finite numbers, it is a jump discontinuity.
- If either $lim_{x \to c^{-}} f (x)$ or $lim_{x \to c^{+}} f (x)$ is $\pm \infty$ , it is an infinite discontinuity.
If Condition 2 holds but Condition 3 fails: The two-sided limit $lim_{x \to c} f (x)$ exists, but $lim_{x \to c} f (x) \neq = f (c)$ . This is the definition of a removable discontinuity. This also covers the case where Condition 1 fails ( $f (c)$ is undefined) but the limit still exists.

Core Concepts & Rules

A discontinuity is removable at $x = c$ if the limit as $x$ approaches $c$ exists. Graphically, this appears as a "hole" in the curve.
A discontinuity is a jump discontinuity at $x = c$ if the left- and right-hand limits both exist but are not equal. This is frequently seen in piecewise-defined functions.
A discontinuity is an infinite discontinuity at $x = c$ if the function's values approach $\infty$ or $- \infty$ as $x$ approaches $c$ from at least one side. This corresponds to a vertical asymptote on the graph.
For a rational function $f (x) = \frac{p ( x )}{q ( x )}$ , if $q (c) = 0$ , look at the factors. If a factor $(x - c)$ in the denominator cancels with a factor in the numerator, the discontinuity at $x = c$ is removable. If the factor does not cancel, the discontinuity is infinite.

Step-by-Step Example 1: Rational Function

Problem: Find and classify all discontinuities for the function $f (x) = \frac{x ^{2} - 9}{x ^{2} + x - 6}$ .

Step 1: Factor the numerator and denominator.

Factoring allows us to analyze the behavior of the function near the points where the denominator is zero.

$f (x) = \frac{( x - 3 ) ( x + 3 )}{( x + 3 ) ( x - 2 )}$

Step 2: Identify potential points of discontinuity.

Discontinuities for a rational function can only occur where the denominator is zero. Set the original denominator to zero: $x^{2} + x - 6 = 0$ , which gives $(x + 3) (x - 2) = 0$ . The potential discontinuities are at $x = - 3$ and $x = 2$ .

Step 3: Classify the discontinuity at $x = - 3$ .

Notice that the factor $(x + 3)$ appears in both the numerator and the denominator. We can cancel these factors to analyze the limit.

$x \to - 3 lim f (x) = x \to - 3 lim \frac{( x - 3 ) ( x + 3 )}{( x + 3 ) ( x - 2 )} = x \to - 3 lim \frac{x - 3}{x - 2}$

Now, substitute $x = - 3$ into the simplified expression:

$x \to - 3 lim \frac{x - 3}{x - 2} = \frac{- 3 - 3}{- 3 - 2} = \frac{- 6}{- 5} = \frac{6}{5}$

Since the limit exists ( $6/5$ ) but the original function is undefined at $x = - 3$ , this is a removable discontinuity.

Step 4: Classify the discontinuity at $x = 2$ .

The factor $(x - 2) F or m u l a [3] lim_{x \to 2} f (x) = lim_{x \to 2} \frac{x - 3}{x - 2} F or m u l a [4] g (x) = ⎩ ⎨ ⎧ 2 x - 1 5 x^{2} x < 2 x = 2 x > 2 F or m u l a [5] lim_{x \to 2^{-}} g (x) = lim_{x \to 2^{-}} (2 x - 1) = 2 (2) - 1 = 3 F or m u l a [6] lim_{x \to 2^{+}} g (x) = lim_{x \to 2^{+}} (x^{2}) = (2)^{2} = 4$ $

Step 4: Compare the limits and classify the discontinuity.

The left-hand limit is 3, and the right-hand limit is 4.

The left-hand limit exists and is finite.
The right-hand limit exists and is finite.
$lim_{x \to 2^{-}} g (x) \neq = lim_{x \to 2^{+}} g (x)$ .

Because the one-sided limits exist but are not equal, the function has a jump discontinuity at $x = 2$ . (Note: The fact that $g (2) = 5$ does not change the classification, which is based on the limits.)

Using Your Calculator

This topic is primarily analytical, meaning you classify discontinuities using limit rules, not calculator functions. However, a graphing calculator is an excellent tool for visualizing the behavior and confirming your analysis.

To visualize discontinuities:

Graph the Function: Enter the function into the Y= editor. For piecewise functions, use the $t es t ‘ m e n u co n d i t i o n s . F ore x am pl e, t h e f u n c t i o n f ro m E x am pl e 2 co u l d b ee n t ere d a s : ‘ Y 1 = (2 X - 1) (X < 2) + (X^{2}) (X > 2) ‘. N o t e t ha tt h ec a l c u l a t orc ann o t g r a p h t h es in g l e p o in t a t$ (2, 5) $.2. * * Vi s u a l I n s p ec t i o n : * * * * * J u m p D i sco n t in u i t y : * * T h e g r a p h w i ll s h o w a c l e a r v er t i c a l b re ak, a s in t h e p i ece w i see x am pl e . * * * I n f ini t eD i sco n t in u i t y : * * T h e g r a p h w i ll s h o w a v er t i c a l a sy m pt o t e, w i t h t h ec u r v e g o in g t o w a r d s p os i t i v eor n e g a t i v e in f ini t yo n e i t h ers i d e . U se ‘ ZOOM ‘ t o g e t ab e tt er v i e w i f n ee d e d .3. * * I n v es t i g a t in g a " Ho l e " (R e m o v ab l eD i sco n t in u i t y) : * * * A re m o v ab l e d i sco n t in u i t y i so f t e nin v i s ib l eo n t h es t an d a r d c a l c u l a t orscree n . T h ec a l c u l a t or ma y d r a w a co n t in u o u s l in e . * T oc h ec k, u se t h e ‘ TR A CE ‘ f e a t u re an d t y p e t h e x - v a l u eo f t h es u s p ec t e dd i sco n t in u i t y (e . g .,$ -3$ for Example 1). If the $y =$ value is blank, but the graph appears connected, you have confirmed a removable discontinuity. You can also use the table (2nd + GRAPH) to see an "ERROR" for the y-value at that specific x-value.

The calculator helps you see the result, but you must use the limit definitions to justify the type of discontinuity on the AP Exam.

AP Exam Quick Hit

Common Question Types

Classifying from a Rational Function: You will be given a function like $f (x) = \frac{p ( x )}{q ( x )}$ and asked to find the location and type of all its discontinuities. This requires factoring.
Classifying from a Piecewise Function: You will be given a piecewise function and asked to determine if it is continuous at the "break" points. If not, you must classify the discontinuity by evaluating the left- and right-hand limits.
Identifying from a Graph: You will be shown a graph with various breaks (holes, jumps, asymptotes) and asked to identify the x-value and type of each discontinuity.

Common Mistakes

Assuming All Zeros in the Denominator are Asymptotes: A common error is to see a value $c$ that makes the denominator zero and immediately classify it as an infinite discontinuity. You must always check if the corresponding factor $(x - c)$ cancels with the numerator, which would indicate a removable discontinuity.
Confusing "Undefined" with "Discontinuous": While a function must be defined at a point to be continuous there, the reason for the discontinuity matters. For a removable discontinuity, the limit exists even if $f (c)$ is undefined. For a jump or infinite discontinuity, the limit does not exist.
Algebraic Errors: Simple mistakes in factoring the numerator or denominator of a rational function will lead to an incorrect classification of the discontinuity.
Incorrectly Evaluating Limits for Piecewise Functions: Students may use the wrong "piece" of the function when evaluating a one-sided limit. Always check if you are approaching from the left ( $x < c$ ) or the right ( $x > c$ ).

Exploring Types of Discontinuities - AP Calculus AB Study Guide