Intuitive Notion of Limits

Introduction

In mathematics, the concept of a limit provides a fundamental way to describe how a function behaves as its input approaches a particular value. Limits are essential in calculus and mathematical analysis, serving as the foundation for defining derivatives, integrals, and continuity. Intuitively, a limit captures the idea of getting arbitrarily close to a desired outcome, even if that outcome is never exactly reached.

To develop an intuitive understanding of limits, consider the example of a car approaching a stop sign. As the car moves closer, its distance to the sign decreases, eventually becoming almost negligible. Although the car never simultaneously occupies the position of the stop sign while still in motion, we can describe its approach using limits.

Definition of a Limit

A function 𝑓 is defined at every number within an open interval containing 𝑎, except possibly at 𝑎 itself. If 𝑓(𝑥) moves closer to a number 𝐿 as 𝑥 approaches 𝑎 from both the left and right, then we say 𝐿 is the limit of 𝑓(𝑥) as 𝑥 nears 𝑎 . This is written as:

To determine the limit of 𝑓(𝑥) as 𝑥 gets closer to 𝑎, we examine the function's values at points near 𝑎 from both directions.

Illustrative Example 1

Consider the function 𝑓 defined by

The attached table below shows the table of values for 𝑓(𝑥), wherein the values of 𝑥 in the table are carefully chosen to show what happens to 𝑓(𝑥) as 𝑥 closer to 4 from both sides—smaller values approaching from the left of 4 and larger values approaching from the right of 4.

Hence, the limit of 𝑓(𝑥) as 𝑥 approaches to 4 is 8. In mathematical terms, we have

With this illustrative example, we cam define the notion of one-sided limits.

One-sided Limits

When evaluating a limit, it is important to consider the direction from which 𝑥 approaches a particular value 𝑎. Specifically, 𝑥 can approach 𝑎 from the left (using values smaller that 𝑎) or from the right (using values greater than 𝑎). This directional approach is indicated by specific notation in limit expressions:

Left-Hand Limit

If 𝑥 approached 𝑎 from the left (through values less than 𝑎), we denote this as

Right-Hand Limit

If 𝑥 approached 𝑎 from the right (through values greater than 𝑎), we denote this as

Moreover, we say

if and only if

Thus, for the limit 𝐿 to exist, the limits from the left and the right must both exist and have the same value.

Illustrative Example 2

Consider the function ℎ defined by ℎ(𝑥) = 3𝑥 − 4. We use the following table to observe the values of ℎ(𝑥) as 𝑥 gets closer and closer to 3 from both sides of 3.

Notice that as 𝑥 get closer and closer to 3 from the left, ℎ(𝑥) approaches 5. The same thing happens to ℎ(𝑥) when 𝑥 gets closer and closer to 3 from the right. Hence, we say that the limit of ℎ(𝑥) as 𝑥 gets closer and closer to 3 is 5, and we write

To fully understand the concept of limit, we will trace 𝑓(𝑥) = 3𝑥 − 4 (the red line) and the line 𝑥 = 3. Looking at the graph of 𝑓(𝑥), we can see that as 𝑥 gets closer to 3 from both the left and the right, the value of 𝑓(𝑥) gets closer to 5. When we trace the line from the left side of 𝑥 = 3, the graph is going up toward the point (3,5), and when we come form the right side, the line also moves toward that same point. Since both sides are approaching the same 𝑦-value, we can say that the limit of 𝑓(𝑥) = 3𝑥 − 4 as 𝑥 approaches 3 is 5. This means that even if 𝑥 isn't exactly 3, the values of the function are getting really close to 5, so the limit is 5.