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Continuity and Limits

This original textbook reading covers part of the AP topic. Follow the AP course outline for complete topic instruction and practice.

This reading includes verified answer corrections and explicitly stated clarifications.

What is a limit?

A limit is the value a function approaches as its input approaches a given point. The function need not reach that value or even be defined at the point. If the function is continuous there, its limit equals its value. Let’s see why this distinction matters.

If the function is continuous:

$lim_{x \to a} f (x) = f (a)$

That is basically the fancy mathematical way of saying “the limit of a given function f(x) as x approaches a (an arbitrary constant value) is equal to the value of the function at point a. However, this is only true if the function is continuous at a! Before we discuss when it is true for discontinuous functions, let’s define continuity.

What is a Continuous Function?

The simplest way to answer this question is this: if you can trace the curve of the function without picking up your pencil or pen, you are dealing with a continuous function. Almost every function you have dealt with so far in studying mathematics starting way back in Algebra 2 has probably been a continuous function.

Gallery 2.1.1 Discontinuities

Read all original image captions

Image 1

Hole Discontinuity - Usually occurs when a function is undefined only at one point.

Image 2

Skip discontinuity - Occurs when a point skips out of what would normally be a continuous function.

Image 3

Jump Discontinuity - Occurs when a part of a function breaks away.

Image 4

This is an interesting combination of a jump and skip discontinuity. We can call it a Hopscotch Discontinuity.

Image 5

Infinite Discontinuity - This occurs when either side of the function approaches infinity.

Image 6

Compression Discontinuity - Although this might not seem like a discontinuity at first glance, if you were to try to trace this function, you would be scratching your pencil on paper until it bore holes through your desk before reaching f(0).

Examples of continuous functions:

Exponential functions
Logarithmic functions
Linear and polynomial functions
Sinusoidal functions

Examples of discontinuous functions:

Rational functions at excluded denominator zeros
Some trigonometric functions (tangent, cotangent, secant, cosecant)
Some piecewise functions
Fractional functions that could have a zero in the denominator

What makes a function discontinuous?

If a function is discontinuous, it is because it contains one or more discontinuities, which can usually be spotted by either observing the graph of the function or its equation. Gallery 2.1.1 shows a few different types of removable and essential (also known as non-removable) discontinuities:

Example Problem 2.1.1.

In the graph of the function below, find:

$lim_{x \to 1^{+}} f (x)$
$lim_{x \to 1^{-}} f (x)$
$lim_{x \to 1} f (x)$

Source figure

Solution:

First let’s decipher what this problem actually means. The first part asks us to “find the limit of the function as x approaches 1 from the right (a good way to remember that + means right is to ask yourself: On which side are all the positive x-values?).” If this was a continuous function, the limit would just be f(1) = 1.5. But this is not a continuous function!

Remember that a limit is the value that any function approaches, so let’s try an exercise to find the limit— if you were to place your finger all the way on the right of the graph, and slide it along the function until it approaches x=1, which value would it take? Try it! Don’t let it touch x=1 though, only just barely let the edge of your finger approach that value... Where is your finger now? It is probably close to the point (1,1). This is the

right hand limit! The limit as x approaches 1 from the right is 1!

This question asks us to “find the limit of the function as x approaches 1 from the left (the negative side).” Since this function is discontinuous, let us try the previous exercise. Drag your finger all the way from the left side of the graph along the line until the edge of it just barely touches x=1. This time, your finger should be close to the point (1, 2). This time, the limit as the function approaches 1 from the left is 2!

The last part asks us to “find the general limit as the function approaches x=1. I am not a fan of theorems but here is a good one to keep in mind when calculating general limits. It is actually two theorems, but they correlate with each other.

Theorem 2.1: If $lim_{x \to a^{+}} f (x) = lim_{x \to a^{-}} f (x)$ , then
$lim_{x \to a^{+}} f (x) = lim_{x \to a^{-}} f (x) = lim_{x \to a} f (x)$

In English, Theorem 2.1 says that if the limit as x approaches a from the left equals the limit as x approaches a from the right, then the general limit as x approaches a is equal to that limit.

Theorem 2.2: If $lim_{x \to a^{-}} f (x) \neq lim_{x \to a^{+}} f (x)$ , then $lim_{x \to a} f (x) DNE$ .

In English, Theorem 2.2 says that if the left hand side and the right hand side limits of a function approaching a value do not equal each other, then the general limit of the function as it approaches that value does not exist!

Using Theorem 2.2, we can see that the answer to part c is... the limit does not exist!

So far we have looked at how to solve limits visually. However, it is also possible to solve them analytically through equations. Just as there are laws in arithmetic and algebra, there are also laws that determine what we can do with limits. Although these laws are very useful, keep in mind that they assume that all limits listed in these laws exist.

Table 2.1 - Limit Laws

Source figure

Table 2.2 - Advanced Limit laws

Source figure *Although written as the Power Law, it also applies as the Root Law by simply making n a fractional number. It even applies for n = 0, which would give 1 as solution to the limit, because anything raised to the 0 power is 1, and the limit as x approaches any value of a constant function is that constant, in this case 1.

**The Composition Law has an additional assumption that f(x) is continuous on the limit of g(x) as x approaches a.

Key point: The interesting thing about limits is that since they only approach a value, we can use them to estimate behaviors of functions that we would not be able to do otherwise. For example, limits allow us to calculate the behavior of a function at infinity and negative infinity, even though all functions are undefined at infinity. How is this possible? It is because limits only approach infinity. They do not ever equal infinity. Practically, this is the same as plugging in a very high or low value for x, such as ±100,000,000, rather than ±infinity.

Squeeze Theorem

The Squeeze Theorem says that if a function can be bounded by an upper bound function and a lower bound function, and the limits of each of those function is known and converges to the same value, then the limit of the bound function also converges to that value. Mathematically: If $lim_{x \to a} f (x)$ is unknown, but there exist functions, g(x) and h(x) such that $g (x) \leq f (x) \leq h (x)$ , and $lim_{x \to a} g (x)$ = $lim_{x \to a} h (x) = L$ , then $lim_{x \to a} f (x) = L$ . In case that was just mumbo jumbo to you, here is a visual explanation.

Review

The 6 types of discontinuities

•Hole discontinuity - Occurs when a function is undefined at a single point.

•Skip discontinuity - Occurs when a continuous function has at least one value that pops out, resulting in an isolated point.

•Jump discontinuity - Occurs when a piece of the function breaks off to create a separate function.

•Hopscotch discontinuity - Occurs when two different piecewise functions have a point that is not common in either of their equation.

•Infinite discontinuity - Occurs for a given x-value when one side of the function approaches +∞ and another side of the function approaches -∞.

•Compression discontinuity - Occurs when a function infinitely oscillates back and forth as it approaches a specific value.

Glossary and deeper reading · 7 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.
Read more: 1.2 · Limit →
Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.
Read more: 1.11 · Continuity →
Polynomial
A polynomial in one variable is a finite sum of constant multiples of nonnegative integer powers of that variable. The degree is the highest exponent with a nonzero coefficient. For example, 2x³−x+5 is a polynomial of degree 3. Expressions containing 1/x or √x are not polynomials in x.
Sinusoidal function
A sinusoidal function has the shape of a sine or cosine graph after shifts, scaling, and possible reflection. A common form is A sin(B(x−h))+D with nonzero A and B. For example, 2 sin(3x)+1 has amplitude 2, midline 1, and period 2π/3. The parameters control vertical variation, repetition, and position of the wave.
Rational function
A rational function is a quotient of two polynomials. Its domain excludes inputs that make the original denominator zero. For example, (x²−1)/(x−1) agrees with x+1 away from x=1, but remains undefined there. Canceling a common factor does not automatically restore an excluded input.
Discontinuity
A discontinuity is a failure of continuity at an input being considered. The obstruction can be a missing or mismatched value, unequal one-sided limits, oscillation, or unbounded behavior. For example, assigning f(0)=5 while f(x)=x elsewhere creates a discontinuity at 0. Identifying the obstruction determines whether changing one value can repair it.
Read more: 1.10 · Discontinuity →