What are limits

Although they may seem daunting due to their notation, limits are actually quite an easy concept. They are essentially the value of a function, say $f(x)$, as it gets closer to a specific value of $x$.

Limit notation is as follows:

$^{\lim}_{x\to a} f(x)$

Where $a$ is the value at which $x$ is approaching, $f(x)$ is the function at which the limit is being applied to, and ($^{\lim}_{x \to a} f(x)$) is "the value of the limit of $f(x)$ when $x$ is approaching $a$".

Moving on, take a look at the following animation:

Notice that as the value of $x$ approaches $0$ (or $x \to 0$) from the left and right, both y values become equal to $1$. This is obvious, since the value of $y = \cos(x)$ is equal to $1$ when $x=0$, however, it is important to consider that a limit is not the value of a function at some $x$ in every case.

Therefore, the limit of $f(x) = \cos(x)$ when x approaches $0$ (or $^{\lim}_{x \to 0} \cos(x) = 1$) is equal to $1$.

example

Determine $\lim_{x \to 2} f(x)$ from the following image:

answer

The limit of the function $f(x)$ as it approaches $x=2$ is equal to $3$, since, from both sides, the function approaches the value $3$.

Left and right limits

There are primarily two types of limits: left and right.

A left limit is the limit of a function when $x$ approaches a value from the left. A right limit is the limit of a function when $x$ approaches a value from a right.

This is important to know, because not all the functions you deal with are continuous. We will discuss more about this in the next lectures, however, take a look at the following example to understand this.

example

Take the following function: $f(x) = \left \{ \begin {matrix} 2 - x \quad \text {for} \ x < 1 \quad (1) \newline x + 1 \quad \text {for} \ x \geq 1 \quad (2) \end{matrix} \right.$

This type of function is called a piecewise function and they are made up of several other functions based on the indicated domains.

The image below depicts what the function above looks like:

For all values $x<1$, the function $(1): \ f(x) = 2 - x$ is used to calculate the $y$ values, while for all values $x \geq 1$, the function $f(x)=2-x \ (2)$ is used to calculate the values of $y$.

Notice that, for $(1)$, there is an open point at $x=1$. This means the function is not defined at that point, since, it is only defined for $x<1$. On the other hand, for $(2)$, the function is defined at $x = 1$, indicated by a closed point since it is defined for $x \geq 1$ (greater than or equal to 1).

Left limit

The left limit of $f(x)$ is indicated by a $(^-)$ superscript. Its value for this example is:

$^{\lim}_{x \to 1^{-}} f(x) = \ ^{\lim}_{x \to 1} (2 - x) = 1$

Note that, even though $(1)$ is not defined at $x=1$, its limit is!

Right limit

The right limit $f(x)$ is indicated by a $(^+)$ superscript. For this case, its value is:

$^{\lim}_{x \to 1^{+}} f(x) = \ ^{\lim}_{x \to 1} (x+1) = 2$

In this example, $^{\lim}_{x \to 1^{-}} f(x) \neq \ ^{\lim}_{x \to 1^{+}} f(x)$, however, there may be cases where they are equal.

Key things to remember

Now, that's all of the basics down, before you go to the next lecture, here are some key tips to remember:

• $^{\lim}_{x \to a} f(x)$ may be defined even though $f(a)$ is not

• $^{\lim}_{x \to a} f(x)$ can be equal to $f(a)$, but not always

• If $L = \ ^{\lim}_{x \to a^{-}} f(x)$ and $L = \ ^{\lim}_{x \to a^{+}} f(x)$, then $L = \ ^{\lim}_{x \to a} f(x)$