Chapter 6: Limits

Limits & Continuity

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.

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 are the foundation for defining derivatives, which describe rates of change.

Example 1: Limits

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

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

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

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

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

if and only if

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

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

Example 2: Limits

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 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 limit, so it does not approach a finite value. We may write $x^{2} \to \infty$ as $x \to \infty$ , but the limit does not exist in the finite sense.

Discontinuity

A discontinuity is a point where the function’s value changes abruptly. In our first example, $f$ has discontinuities at $x = 1$ and $x = π$ because the limits at these points do not exist.

Example: ...

Consider the function shown below, where $f$ has jumps at $x = 1$ and $x = π$ .

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

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

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 $x = π$ .

However, discontinuity does not always mean the limit fails to exist. For instance, consider the following example.

Example 3: Limits

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

Looking at the graph, it is clear that despite $lim_{x \to 0} f (x) = 2$ , the actual function value is $f (0) = 1$ . In this case, we say that $f$ is discontinuous at $x = 0$ .

Continuity

The discussion about when limits exist and do not exist, leads us to the following definition.

Definition: Continuity

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

The function $f$ is said to be continuous at a point $c \in D$ 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 $c$ .

Breaking 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 $D$ .
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 $c \in D$ , then we say that $f$ is continuous on $D$ .

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: ...

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 $x = 0$ .

Limit Laws

In an earlier chapter, we saw how to combine two functions using arithmetic operations. The table below shows the corresponding rules for limits, assuming both $x \to a lim f (x)$ and $x \to a lim g (x)$ exist.

Table 1. ...

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

Example 1: ...

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 2: ...

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:

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

Evaluate the expression (direct subsitution):

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

Mathematics Brush-up for Data Science

Chapter 6: Limits

Limits & Continuity

Discontinuity

Continuity

Limit Laws

Chapter Exercise

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Mathematics Brush-up for Data Science

Chapter 6: Limits

Limits & Continuity

Discontinuity

Continuity

Limit Laws

Chapter Exercise