Introduction to Limits

The concept of limit is one of the fundamental ideas of Differential and Integral Calculus.

It allows us to study the behavior of a function when the independent variable approaches a given value.

The central idea is simple:

The limit describes the value a function approaches as $x$ approaches a given number.

1. The Intuitive Idea of a Limit

Consider the function

$$f(x)=2x+1.$$

We want to determine what happens to $f(x)$ as $x$ approaches $3$.

We can calculate some values close to $3$:

$x$	$f(x)=2x+1$
$2$	$5$
$2{,}9$	$6{,}8$
$2{,}99$	$6{,}98$
$2{,}999$	$6{,}998$
$3$	$7$
$3{,}001$	$7{,}002$
$3{,}01$	$7{,}02$
$3{,}1$	$7{,}2$
$4$	$9$

We observe that, as $x$ approaches $3$, the values of $f(x)$ approach $7$.

We write:

$$\boxed{\lim_{x\to3}f(x)=7}$$

or, substituting $f(x)=2x+1$,

$$\boxed{\lim_{x\to3}(2x+1)=7}.$$

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2. What Does $x\to a$ Mean?

The notation

$$x\to a$$

means that $x$ is approaching $a$.

This does not necessarily mean that $x=a$.

For example, when we write

$$x\to2,$$

we can have values such as

$$1{,}9,\quad 1{,}99,\quad 1{,}999,\quad 2{,}001,\quad 2{,}01,\quad 2{,}1.$$

All these values are close to $2$, but they are different from $2$.

This distinction will be very important in the study of limits.

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3. The Meaning of the Expression $\lim_{x\to a}f(x)$

The expression

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

should be read as:

When $x$ approaches $a$, the values of $f(x)$ approach $L$.

For example,

$$\lim_{x\to2}(x^2+1)=5.$$

This means that, as $x$ approaches $2$, the value of $x^2+1$ approaches $5$.

Notice that we are interested in the behavior of the function near $2$.

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4. The Limit Is Not Necessarily the Value of the Function

One of the most important points in the study of limits is understanding that

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

and

$$f(a)$$

are different concepts.

Consider, for example,

$$f(x)= \begin{cases} x+1, & x\neq2,\\ 10, & x=2. \end{cases}$$

For values of $x$ close to $2$, but different from $2$, we have

$$f(x)=x+1.$$

Therefore,

$$\lim_{x\to2}f(x)=3.$$

However,

$$f(2)=10.$$

Therefore,

$$\boxed{\lim_{x\to2}f(x)=3}$$

while

$$\boxed{f(2)=10}.$$

The limit depends on the behavior of the function near $2$, and not necessarily on the value assigned to the function exactly at $2$.

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5. Approaching from the Left and from the Right

When $x$ approaches a number $a$, we can approach it in two ways:

• from the left;
• from the right.

From the Left

We write

$$x\to a^-$$

and this means that $x$ approaches $a$ by taking values less than $a$.

For example:

$$1{,}9,\quad1{,}99,\quad1{,}999,\ldots$$

when $a=2$.

The corresponding limit is called the left-hand limit:

$$\lim_{x\to a^-}f(x).$$

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From the Right

We write

$$x\to a^+$$

and this means that $x$ approaches $a$ by taking values greater than $a$.

For example:

$$2{,}1,\quad2{,}01,\quad2{,}001,\ldots$$

when $a=2$.

The corresponding limit is called the right-hand limit:

$$\lim_{x\to a^+}f(x).$$

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6. When Does the Limit Exist?

For the limit

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

to exist, the one-sided limits must exist and be equal.

That is,

$$\boxed{ \lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) }$$

In this case, we can write

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

Example

Consider a function whose behavior near $2$ is such that

$$\lim_{x\to2^-}f(x)=5$$

and

$$\lim_{x\to2^+}f(x)=5.$$

Since the two one-sided limits are equal, we have

$$\boxed{\lim_{x\to2}f(x)=5}.$$

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7. When Does the Limit Not Exist?

Now consider a function for which

$$\lim_{x\to2^-}f(x)=3$$

and

$$\lim_{x\to2^+}f(x)=7.$$

Since

$$3\neq7,$$

the one-sided limits are different.

Therefore,

$$\boxed{\lim_{x\to2}f(x)\text{ does not exist}.}$$

This situation is very important in the study of continuity.

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8. Calculating Limits by Direct Substitution

For many functions, we can calculate the limit simply by substituting the value that $x$ is approaching.

For example:

$$\lim_{x\to3}(x^2+2x+1).$$

Substituting $x=3$:

$$3^2+2(3)+1.$$

Therefore,

$$9+6+1=16.$$

Thus,

$$\boxed{ \lim_{x\to3}(x^2+2x+1)=16 }.$$

This happens because polynomials are continuous functions for all real numbers.

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9. An Example Where Direct Substitution Fails

Consider

$$\lim_{x\to2}\frac{x^2-4}{x-2}.$$

If we substitute $x=2$ directly, we obtain

$$\frac{2^2-4}{2-2} = \frac00.$$

The expression $\frac00$ is an indeterminate form.

This does not mean that the limit does not exist.

We need to manipulate the expression before calculating the limit.

Factoring the numerator:

$$x^2-4=(x-2)(x+2).$$

Thus,

$$\frac{x^2-4}{x-2} = \frac{(x-2)(x+2)}{x-2}.$$

For $x\neq2$, we can simplify:

$$\frac{x^2-4}{x-2}=x+2.$$

Therefore,

$$\lim_{x\to2}\frac{x^2-4}{x-2} = \lim_{x\to2}(x+2).$$

Now we can use direct substitution:

$$2+2=4.$$

Thus,

$$\boxed{ \lim_{x\to2}\frac{x^2-4}{x-2}=4 }.$$

This example shows a fundamental idea:

An indeterminate form such as $\frac00$ is not the result of the limit. It indicates that we need to perform some algebraic procedure before determining the value of the limit.

10. Limits and the Graph of a Function

The concept of a limit can also be understood geometrically.

When we write

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

we are saying that, as we observe the graph of the function increasingly close to the vertical line $x=a$, the values of $y$ approach $L$.

Visually, we can think of the point

$$(a,L)$$

as indicating where the graph is heading as $x$ approaches $a$.

This remains true even if the point $(a,L)$ does not belong to the graph.

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11. Limits at Infinity

We can also study the behavior of a function as $x$ grows indefinitely.

In this case, we use the notation

$$x\to+\infty.$$

For example,

$$\lim_{x\to+\infty}\frac1x=0.$$

This means that, as $x$ grows larger and larger, the value of $\frac1x$ gets closer and closer to $0$.

Similarly,

$$\lim_{x\to-\infty}\frac1x=0.$$

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12. Why Are Limits Important?

The concept of a limit is the foundation for several fundamental concepts in Calculus.

From limits, we can develop:

• continuity;
• derivatives;
• instantaneous rates of change;
• integrals;
• areas and volumes;
• series and approximations.

For example, the derivative of a function can be defined using the limit

$$f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}{h}.$$

Therefore, mastering limits is essential for progressing in the study of Calculus I.

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13. Summary

The fundamental concept can be summarized as follows:

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

means that the values of $f(x)$ approach $L$ as $x$ approaches $a$.

We should remember that:

1. $x\to a$ means that $x$ approaches $a$;
2. the limit does not necessarily depend on $f(a)$;
3. the left-hand limit is represented by $x\to a^-$;
4. the right-hand limit is represented by $x\to a^+$;
5. the limit exists when the one-sided limits are equal;
6. direct substitution works in many cases;
7. expressions such as $\frac00$ are indeterminate forms and require further analysis;
8. limits are the foundation for the study of derivatives and continuity.

In the next topic, we can study the properties of limits and the main techniques for calculating limits, including factoring, rationalization, and one-sided limits.