2.2 The Limit of a Function

Understanding Limits

Concept and notation of limits

A limit describes the value a function approaches as the input gets arbitrarily close to a specific value, without necessarily reaching it. The key idea: you care about what happens near a point, not at the point.

The notation looks like this:

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

This reads as "the limit of $f(x)$ as $x$ approaches $a$ equals $L$." It means that as $x$ gets closer and closer to $a$ (from either side), the output $f(x)$ gets closer and closer to $L$.

Why does this matter? Sometimes a function isn't even defined at $x = a$ (think of a hole in a graph), but the limit can still exist. The limit tells you where the function is headed, regardless of whether it actually arrives.

Limit estimation techniques

There are two main ways to estimate a limit before you learn algebraic techniques.

Using a table of values:

1. Pick $x$-values that approach your target from the left (smaller values) and from the right (larger values). For example, if you're finding $\lim_{x \to 2} f(x)$, try $x = 1.9, 1.99, 1.999$ and $x = 2.1, 2.01, 2.001$.
2. Calculate $f(x)$ for each of those values.
3. Look at whether the outputs are converging toward a single number. If they are, that number is your estimated limit.

Using a graph:

• Trace the curve from both the left and right sides toward the point of interest.
• If both sides approach the same $y$-value, that's the estimated limit.
• Watch for holes, jumps, or vertical asymptotes at the point. These signal that the limit might not exist or that the one-sided limits differ.

Types of Limits and Their Properties

One-sided vs. two-sided limits

A two-sided limit asks what $f(x)$ approaches as $x$ comes in from both directions. A two-sided limit exists only when the left-hand and right-hand limits are equal.

One-sided limits restrict the approach to just one direction:

• Left-hand limit: $\lim_{x \to a^-} f(x)$ means $x$ approaches $a$ from values less than $a$.
• Right-hand limit: $\lim_{x \to a^+} f(x)$ means $x$ approaches $a$ from values greater than $a$.

The connection between them is straightforward:

$\lim_{x \to a} f(x) = L \quad \text{if and only if} \quad \lim_{x \to a^-} f(x) = L \quad \text{and} \quad \lim_{x \to a^+} f(x) = L$

If the two one-sided limits give different values, the two-sided limit does not exist.

Non-existent limits

A limit fails to exist in several situations:

• Jump discontinuity: The left-hand and right-hand limits exist but aren't equal. For example, a piecewise function that jumps from 3 to 5 at $x = 1$.
• Vertical asymptote: The function blows up toward $\infty$ or $-\infty$ near the point.
• Oscillation: The function bounces back and forth without settling down. A classic example is $f(x) = \sin\!\left(\frac{1}{x}\right)$ near $x = 0$, which oscillates infinitely fast.

Infinite limits and vertical asymptotes

An infinite limit means the function grows without bound as $x$ approaches a particular value:

• $\lim_{x \to a} f(x) = \infty$ means $f(x)$ increases without bound.
• $\lim_{x \to a} f(x) = -\infty$ means $f(x)$ decreases without bound.

Technically, when we write $\lim_{x \to a} f(x) = \infty$, the limit does not exist in the usual sense ($\infty$ is not a real number). The notation is a shorthand describing how the limit fails to exist.

A vertical asymptote occurs at $x = a$ whenever at least one of the one-sided limits is $\infty$ or $-\infty$. For example, $f(x) = \frac{1}{x - 3}$ has a vertical asymptote at $x = 3$ because $\lim_{x \to 3^+} \frac{1}{x-3} = \infty$ and $\lim_{x \to 3^-} \frac{1}{x-3} = -\infty$.

Applications of Limits

Real-world applications of limits

The most important application you'll see in Calculus I is instantaneous velocity. Average velocity over a time interval is $\frac{\Delta \text{position}}{\Delta \text{time}}$. As you shrink that time interval toward zero, the limit of the average velocity gives you the velocity at a single instant. This is exactly how derivatives are built.

Other applications include:

• Modeling how a physical quantity (like current or temperature) behaves as a variable approaches a critical threshold
• Analyzing what happens to a cost or revenue function as production approaches a certain level

Function behavior near critical points

Limits let you classify what's happening at tricky spots on a graph:

• If $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$, the function is continuous at $x = a$. No gaps, no jumps, no surprises.
• If the one-sided limits both exist but aren't equal, there's a jump discontinuity.
• If one or both one-sided limits are infinite, there's a vertical asymptote.
• If the limit exists but doesn't equal $f(a)$ (or $f(a)$ is undefined), there's a removable discontinuity (a hole in the graph).

Function Properties and Limits

Continuity and differentiability

Continuity at a point $x = a$ requires three things:

1. $f(a)$ is defined.
2. $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$.

If any one of these fails, the function is discontinuous at that point.

Differentiability goes a step further. A function is differentiable at $x = a$ if the following limit exists:

$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

This is the difference quotient, and its limit is the derivative. You'll work with this extensively in the next unit. For now, the takeaway is that limits are the machinery that makes derivatives possible.

One useful fact: if a function is differentiable at a point, it's automatically continuous there. But the reverse isn't always true (think of a sharp corner, like $f(x) = |x|$ at $x = 0$, which is continuous but not differentiable).

Domain, range, and asymptotes

Limits connect to the bigger picture of how a function is shaped:

• Domain: Limits help you understand what happens at points excluded from the domain. For instance, $f(x) = \frac{1}{x}$ isn't defined at $x = 0$, but the limit behavior there tells you about the vertical asymptote.
• Range: Horizontal asymptotes can reveal bounds on the range. These are found using limits at infinity: $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$. (You'll cover limits at infinity in more detail soon.)
• Vertical asymptotes are identified by infinite limits as $x$ approaches a finite value.
• Horizontal asymptotes are identified by finite limits as $x$ approaches $\infty$ or $-\infty$.