Pre-Calculus: Limits and Continuity Introduction

You are standing at the edge of calculus, and the bridge to cross over is built from the concepts of limits and continuity. These ideas form the bedrock upon which derivatives and integrals—the twin pillars of calculus—are constructed. Understanding limits allows you to describe behavior that is approaching, but not necessarily reaching, a specific point, which is essential for defining instantaneous rates of change and the area under a curve.

What is a Limit? Graphical and Numerical Intuition

Informally, the limit of a function describes the value that the function's output ($f(x)$) approaches as the input ($x$) approaches a certain number. The crucial word is approaches. We are not concerned with the function's value at the point, but rather the trend of its values near that point.

You can develop this intuition in two primary ways: graphically and numerically. Graphically, you imagine placing your finger on the graph and tracing it toward the $x$-value in question from both the left and right sides. What $y$-value are you heading toward? Numerically, you create a table of values, choosing $x$-values that get progressively closer to the target number from both sides and observing the corresponding $f(x)$ values. If the $y$-values from the left and right converge to the same number, that number is the limit.

For example, consider the limit of $f(x)=\frac{x^2-1}{x-1}$ as $x$ approaches 1. The function is undefined at $x=1$ because it leads to division by zero. However, by examining values close to 1, a pattern emerges:

• From the left ($x=0.9,0.99,0.999$): $f(x)=1.9,1.99,1.999$
• From the right ($x=1.1,1.01,1.001$): $f(x)=2.1,2.01,2.001$

Both sides appear to be approaching the value 2. We conclude that ${\lim }_{x\to 1}f(x)=2$, even though $f(1)$ itself does not exist.

One-Sided Limits and When Limits Fail to Exist

Sometimes, the behavior of a function as it approaches a point differs depending on the direction of approach. This is where one-sided limits are essential. The notation ${\lim }_{x\to a^{-}}f(x)$ denotes the limit as $x$ approaches $a$ from the left (using values less than $a$). Conversely, ${\lim }_{x\to a^{+}}f(x)$ denotes the limit from the right.

For the two-sided limit ${\lim }_{x\to a}f(x)$ to exist, the left-hand and right-hand limits must both exist and be equal: $$\underset{x\to a}{\lim }f(x)=L\text{if and only if}\underset{x\to a^{-}}{\lim }f(x)=L\text{and}\underset{x\to a^{+}}{\lim }f(x)=L.$$

A limit fails to exist in several common scenarios:

1. The one-sided limits are not equal. This is often seen in piecewise functions or functions with a jump.
2. Unbounded behavior. As $x$ approaches $a$, $f(x)$ increases or decreases without bound (e.g., a vertical asymptote).
3. Oscillating behavior. The function oscillates infinitely often between values near a point and never settles down (e.g., $\sin (1/x)$ as $x$ approaches 0).

Types of Discontinuities

A function is continuous at a point $x=a$ if three conditions are met: (1) $f(a)$ is defined, (2) ${\lim }_{x\to a}f(x)$ exists, and (3) ${\lim }_{x\to a}f(x)=f(a)$. A discontinuity occurs when any of these conditions fails. We classify them into main types:

• Removable Discontinuity (Point Discontinuity): This occurs when the limit ${\lim }_{x\to a}f(x)$ exists, but is not equal to $f(a)$ (either because $f(a)$ is a different value or is undefined). Graphically, it looks like a hole in the graph. The limit from our first example, ${\lim }_{x\to 1}\frac{x^2-1}{x-1}=2$, showcases a removable discontinuity at $x=1$.
• Jump Discontinuity: This happens when the one-sided limits exist but are not equal (${\lim }_{x\to a^{-}}f(x)\ne {\lim }_{x\to a^{+}}f(x)$). The graph "jumps" from one value to another. A classic example is the greatest integer function, $f(x)=\lfloor x\rfloor$, at any integer value.
• Infinite Discontinuity: The function exhibits unbounded behavior as $x$ approaches $a$, typically due to a vertical asymptote. For instance, $f(x)=1/x$ has an infinite discontinuity at $x=0$, as the limits from the left and right go to $-\infty$ and $+\infty$, respectively.

Applying Limit Laws for Algebraic Evaluation

While graphical and numerical methods build intuition, limit laws provide the algebraic tools for rigorous evaluation. These laws are theorems that allow you to break down complex limits into simpler parts. Assume ${\lim }_{x\to a}f(x)$ and ${\lim }_{x\to a}g(x)$ both exist. Key laws include:

• Sum/Difference Law: ${\lim }_{x\to a}[f(x)\pm g(x)]={\lim }_{x\to a}f(x)\pm {\lim }_{x\to a}g(x)$
• Constant Multiple Law: ${\lim }_{x\to a}[c\cdot f(x)]=c\cdot {\lim }_{x\to a}f(x)$
• Product Law: ${\lim }_{x\to a}[f(x)\cdot g(x)]={\lim }_{x\to a}f(x)\cdot {\lim }_{x\to a}g(x)$
• Quotient Law: ${\lim }_{x\to a}[f(x)/g(x)]=\frac{{\lim }_{x\to a}f(x)}{{\lim }_{x\to a}g(x)}$, provided ${\lim }_{x\to a}g(x)\ne 0$
• Power Law: ${\lim }_{x\to a}[f(x){]}^n=[{\lim }_{x\to a}f(x){]}^n$

These laws lead to the powerful technique of direct substitution. If a function is created from basic functions (polynomials, rationals, roots, etc.) and is continuous at $a$, then ${\lim }_{x\to a}f(x)=f(a)$. For expressions that yield an indeterminate form like $0/0$ (as in our first example), algebraic manipulation—such as factoring, expanding, or rationalizing—is used to rewrite the function into a form where direct substitution can be applied.

Common Pitfalls

1. Confusing the limit value with the function value. The most critical mistake is assuming ${\lim }_{x\to a}f(x)$ must equal $f(a)$. Remember, the limit describes the journey of $x$ toward $a$, not the destination at $a$ itself. A function can have a limit at a point where it is not even defined.
2. Assuming two-sided existence from one direction. You cannot conclude a limit exists by only checking values from the left or the right. Always verify both one-sided limits. A graph that seems to approach a value from one side might make a sudden jump from the other.
3. Misapplying limit laws to forms that are not indeterminate. The limit laws require the individual limits to exist. You cannot use them to "split" a limit like ${\lim }_{x\to 0}(1/x)$ into a quotient of limits, because ${\lim }_{x\to 0}1$ and ${\lim }_{x\to 0}x$ exist, but the denominator's limit is 0, violating the Quotient Law's condition.
4. Overlooking holes when determining continuity. A graph that looks continuous except for a single tiny hole is, by the formal definition, discontinuous at that point. Continuity requires the function value to equal the limit, not just for the limit to exist.

Summary

• The limit ${\lim }_{x\to a}f(x)=L$ means the output $f(x)$ gets arbitrarily close to $L$ as $x$ gets close to (but not equal to) $a$. Intuition is built through graphs and tables of values.
• The two-sided limit exists only if the one-sided limits from the left and right both exist and are equal. Limits fail to exist due to jumps, unbounded behavior, or oscillation.
• A function is continuous at a point if the limit exists, the function is defined there, and the two values are equal. Breaks in continuity are classified as removable discontinuities (holes), jump discontinuities, or infinite discontinuities (vertical asymptotes).
• Limit laws provide the algebraic rules for evaluating limits, which often lead to the technique of direct substitution for continuous functions. Indeterminate forms require algebraic simplification first.
• Mastering these concepts provides the essential foundation for calculus, where limits are used to define the derivative (instantaneous rate of change) and the integral (accumulation of area).