Limits and continuity

In analysis, much of the work is about describing how functions behave when approaching certain points, or when x grows towards infinity. To talk about this, we use the concepts of limit and continuity.

 

 

What is a limit?

A limit describes the value that a function approaches as x gets closer and closer to a certain point. Even if the function does not necessarily have a value exactly at that point, the limit may still exist.

 

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

 

This means that as x approaches a, the value of the function approaches the number L.

 

We can also speak of left-hand and right-hand limits, which look at what happens when approaching a from each side. If the two limits are equal, we say that the function has a finite limit at that point.

 

Limits are used to describe the behavior near jumps, holes or infinite growths in functions. They therefore play a central role throughout differential and integral calculus.

 

 

Examples of limits

A simple example is the function \( \large f(x) = 2x + 1 \). When x approaches 3, we can calculate the limit directly:

 

$$ \large \lim_{x \to 3} (2x + 1) = 2 \cdot 3 + 1 = 7 $$

 

This means that the function approaches the value 7 as x approaches 3. Here, the function is also defined at that point, so the actual value and the limit are the same.

 

But sometimes the function is not defined at the point, even though the limit exists. Consider for example the function

 

$$ \large f(x) = \frac{x^2 - 9}{x - 3} $$

 

We cannot substitute x = 3 directly, since the denominator becomes zero. But if we simplify the expression, we get

 

$$ \large f(x) = x + 3 \quad \text{for } x \neq 3 $$

 

and therefore

 

$$ \large \lim_{x \to 3} \frac{x^2 - 9}{x - 3} = 6 $$

 

Even though the function is not defined at x = 3, the limit still exists. Graphically, this corresponds to a small “hole” in the graph at the point (3, 6).

 

 

Limits at infinity

We can also study how a function behaves when x grows without bound. For example:

 

$$ \large \lim_{x \to \infty} \frac{1}{x} = 0 $$

 

This means that \( \frac{1}{x} \) becomes smaller and smaller as x increases, eventually approaching zero. In the same way, we can describe functions that grow towards infinity.

 

 

Continuity

A function is continuous at a point if there is no “jump” in the graph at that point. More precisely, the limit and the actual value of the function must be equal:

 

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

 

This means that the graph can be drawn without lifting the pencil. If the limit does not exist, or if the function skips a point, the function is not continuous there.

 

Continuity is an important property because it ensures that the function changes smoothly. Many theorems in analysis — such as the intermediate value theorem and differentiability — require that the function is continuous.

 

 

Importance in analysis

Limits and continuity form the foundation of all analysis. They are prerequisites for defining the derivative, which describes rates and slopes, and later the integral, which describes areas and accumulations.

 

In the following topics, we will see how limits can be calculated systematically using rules, and how to analyze special situations where limits lead to infinite or undefined results.

 

By understanding both limits and continuity, one gains a clear picture of how functions behave, and thus the basis for all further analysis.