Chapter 9: Limits

Limits describe the behavior of functions near a point and provide the foundation for continuity and differentiation. This chapter introduces limits, uses them to define continuity and discontinuity, and then presents the main limit laws used in calculations.

Limits

The concept of a limit concerns the value that a function approaches as its input gets closer to a particular point. Importantly, what happens exactly at that point is not what matters, instead, we focus on what happens around the point.

Function values approaching a limit as x approaches c — Figure 9.1. Illustration of the limit idea: as $x$ approaches $c$ , the corresponding values $f (x)$ move toward a height $L$ . This limiting height need not equal $f (c)$ .

Understanding the concept of a limit is fundamental as it allows us to study the continuity of functions, a property describing whether a function behaves smoothly without abrupt jumps or breaks. Furthermore, limits provide the foundation for the derivatives introduced in Chapter 10, which describe rates of change.

Example: A Limit from a Graph

To illustrate this and build an intuitive understanding of the concept of limit, consider the function and its corresponding graph:

$f (x) = ⎩ ⎨ ⎧ x^{2}, 2, sin (x), x \in [0, 1) x \in [1, π) x \in [π, 2 π]$

Graph of a piecewise function with jumps at one and pi. — Figure 9.2. Piecewise function used to illustrate limits at ordinary points and jump points.

Let us consider the limit of $f (x)$ as $x \to \frac{1}{2}$ . This means that we look at values of $f (x)$ for $x$ in some small interval around $\frac{1}{2}$ . In this case, this corresponds to examining the function $x^{2}$ on the interval $[0, 1)$ . If we look the graph of this function, we can see that as $x$ approaches $\frac{1}{2}$ from either the left or the right, the value of $f (x)$ approaches $\frac{1}{4}$ .

Therefore, the limit of $f (x)$ as $x \to \frac{1}{2}$ can be expressed symbolically as:

$x \to \frac{1}{2} lim f (x) = \frac{1}{4}$

This example illustrates the basic idea behind limits. We now state this idea in more general terms.

Definition: Limit

Let $f (x)$ be a function defined on an open interval containing $c$ (with the possible exception of $c$ itself). Let $L$ be a real number. If all values of the function $f (x)$ approach the real number $L$ as the values of $x \neq = c$ approach the number $c$ , then we say that the limit of $f (x)$ as $x$ approaches $c$ is $L$ .

In words, as $x$ gets closer to $c$ , $f (x)$ gets closer and stays close to $L$ . Symbolically, we express this idea as

$x \to c lim f (x) = L$

if and only if

$x \to c^{+} lim f (x) = L and x \to c^{-} lim f (x) = L .$

Here $x \to c^{+}$ means approaching $c$ from the right (positive direction) and $x \to c^{-}$ means approaching from the left (negative direction).

Example: When a Limit Does Not Exist

Not every function has a limit at every point. A limit may fail to exist for several reasons. To illustrate this, we return to our earlier example and now consider $x = 1$ and $x = π$ . At each of these points, the left-hand and right-hand limits differ:

$L^{-} = x \to π^{-} lim f (x) = 2 and L^{+} = x \to π^{+} lim f (x) = 0.$

Since these one-sided limits are not equal, $x \to π lim f (x)$ does not exist. The same reasoning applies at $x = 1$ .

Another way a finite limit can fail to exist is if the function grows without bound. For example,

$x \to \infty lim x^{2}$

As $x$ increases, $x^{2}$ grows without bound. We express this behavior using the extended notation

$x \to \infty lim x^{2} = \infty.$

Here $x \to \infty$ describes the input becoming arbitrarily large, while the symbol $\infty$ on the right describes the output growing without bound. The symbol $\infty$ is not a real number, so this does not represent a finite limit.

Limits at Infinity and Infinite Limits

The notation

$x \to \infty lim f (x) = L$

means that $f (x)$ approaches the finite value $L$ as $x$ becomes arbitrarily large. For example,

$x \to \infty lim \frac{1}{x} = 0.$

By contrast, writing $lim_{x \to c} f (x) = \infty$ means that the outputs grow without bound as $x$ approaches the finite number $c$ . One-sided behavior still matters. For example,

$x \to 0^{+} lim \frac{1}{x} = \infty and x \to 0^{-} lim \frac{1}{x} = - \infty.$

Because these one-sided behaviors do not agree, the two-sided limit $lim_{x \to 0} \frac{1}{x}$ does not exist.

Continuity

The discussion about when limits exist and do not exist leads to the concept of continuity. A continuous function has no jump, gap, or mismatch between the value approached by the function and the value the function actually takes.

Definition: Continuity

Let $f : A \to R$ be a function, where $A \subseteq R$ is the domain of $f$ .

At an interior point $c$ of $A$ , the function $f$ is said to be continuous at $c$ if

$x \to c lim f (x) = f (c)$

That is, as $x$ approaches $c$ , the value of $f (x)$ approaches the same number that $f$ actually takes at $c$ .

If $c$ is an endpoint of the domain, continuity is defined using the one-sided limit from within the domain. For example, at a left endpoint $a$ , continuity means

$x \to a^{+} lim f (x) = f (a) .$

At an interior point, we can break this definition into its essential parts:

The limit exists: The left-hand and right-hand limits of $f (x)$ at $c$ are equal. In symbols,

$x \to c^{-} lim f (x) = x \to c^{+} lim f (x) = L$

This ensures that $f (x)$ approaches a single, well-defined value near $c$ .
The function value is defined: The point $f (c)$ exists, meaning that $c$ belongs to the domain $A$ . Without a defined value at $c$ , the concept of continuity cannot apply.
The limit equals the function value: The value that $f (x)$ approaches near $c$ is exactly the same as the value it takes at $c$ :

$L = f (c)$

This guarantees there is no jump or gap in the function’s behavior at that point.

If $f$ is continuous at every point $c \in A$ , then we say that $f$ is continuous on $A$ .

Continuity can therefore be viewed as a local property at each point, which extends to a global property when it holds throughout the entire domain.

In practice, most functions we encounter (such as polynomial, exponential, logarithmic, and trigonometric functions) are continuous on the domains we care about.

Example: Checking Continuity at a Point

Consider the function

$f (x) = {- x^{2} + 2, 1, x \in R ∖ {0} x = 0$

We want to determine whether $f$ is continuous at $x = 0$ .

The limit exists:

$x \to 0^{+} lim f (x) = 2 and x \to 0^{-} lim f (x) = 2$

Since both one-sided limits are equal, the limit $lim_{x \to 0} f (x)$ exists and equals 2. The condition is satisfied.
The function value is defined:

$f (0) = 1$

The function has a defined value at $x = 0$ . The condition is satisfied.
The limit equals the function value:

$x \to 0 lim f (x) = 2 \neq = f (0) = 1$

The limit and the function value are not equal. The condition is not satisfied.

Since the third condition fails, $f$ is not continuous at $x = 0$ .

Discontinuity

Continuity gives us a precise way to describe when a function behaves smoothly at a point. A discontinuity occurs when at least one part of the continuity definition fails.

Definition: Discontinuity

A function $f$ is discontinuous at a point $c$ if it is not continuous at $c$ .

This can happen if:

the two-sided limit $lim_{x \to c} f (x)$ does not exist,
the function value $f (c)$ is not defined, or
the limit exists but does not equal the function value.

Example: Jump Discontinuity

Consider the function from the earlier limit example, where $f$ has jumps at $x = 1$ and $x = π$ .

At $x = π$ , the one-sided limits are different:

$x \to π^{-} lim f (x) = 2 and x \to π^{+} lim f (x) = 0.$

Because the left-hand and right-hand limits do not agree, the two-sided limit $lim_{x \to π} f (x)$ does not exist. Therefore, $f$ has a jump discontinuity at $x = π$ .

Discontinuity does not always mean that the limit fails to exist. Sometimes the limit exists, but the function value is different.

Example: Discontinuity with an Existing Limit

Here is an example where the graph approaches one value, but the function is defined to take another value at the point.

$f (x) = {- x^{2} + 2, 1, x \in R ∖ {0} x = 0$

Graph with an open point at zero and a separate filled point showing a removable discontinuity. — Figure 9.3. Removable discontinuity at $x = 0$ : the limit is $2$ while $f (0) = 1$ .

Looking at the graph, it is clear that despite $lim_{x \to 0} f (x) = 2$ , the actual function value is $f (0) = 1$ . Since the limit and function value are not equal, $f$ is discontinuous at $x = 0$ .

Limit Laws

In Chapter 5, arithmetic operations were used to combine two functions. The table below shows the corresponding rules for limits, assuming both $x \to c lim f (x)$ and $x \to c lim g (x)$ exist.

Rules: Limit Laws

The following rules allow us to compute limits of combined functions from the limits of the individual functions.

Table 9.1. Limit laws for combining functions.

Operation	Limit Law
Constant Multiple	$x \to c lim [k \cdot f (x)] = k \cdot x \to c lim f (x)$
Sum/Difference	$x \to c lim [f (x) \pm g (x)] = x \to c lim f (x) \pm x \to c lim g (x)$
Product	$x \to c lim [f (x) \cdot g (x)] = x \to c lim f (x) \cdot x \to c lim g (x)$
Quotient	$x \to c lim \frac{f ( x )}{g ( x )} = \frac{lim _{x \to c} f ( x )}{lim _{x \to c} g ( x )}, x \to c lim g (x) \neq = 0$

Rule: Direct Substitution

If $f$ is continuous at $c$ , then

$x \to c lim f (x) = f (c) .$

Thus, direct substitution can be used for polynomials and for rational, root, exponential, logarithmic, and trigonometric expressions at points where the expression is defined and continuous.

Example: Using Limit Laws

This calculation shows how the limit laws let us split a combined expression into simpler parts.

Determine $lim_{x \to 1} (x + 3 - 4 - x)$

Split into two limits (use the difference law):

$x \to 1 lim x + 3 - x \to 1 lim 4 - x$

Evaluate each part directly (direct substitution):

$x \to 1 lim x + 3 - x \to 1 lim 4 - x = 1 + 3 - 4 - 1 = 4 - 3 = 2 - 3$

Example: Factoring Before Taking a Limit

Sometimes direct substitution produces an indeterminate form; then algebraic simplification can reveal the limit.

Determine $lim_{x \to 1} \frac{x ^{2} - 1}{x - 1}$

Check direct substitution (does not work!):

$\frac{1 ^{2} - 1}{1 - 1} = \frac{0}{0}$

Factor numerator and cancel the common factor. For $x \neq = 1$ , the fraction simplifies to $x + 1$ . Although the original function is not defined at $x = 1$ , the limit only concerns nearby values with $x \neq = 1$ :

$x \to 1 lim \frac{x ^{2} - 1}{x - 1} = x \to 1 lim \frac{( x - 1 ) ( x + 1 )}{x - 1} = x \to 1 lim x + 1$

Evaluate the expression (direct substitution):

$x \to 1 lim x + 1 = 1 + 1 = 2$