Introduction to Limits in Calculus

Numerical and graphical approaches are used to introduce to the concept of limits using examples.

Numerical Approach to Limits

Example 1

Let $f(x) = 2x + 2$ and compute $f(x)$ as $x$ takes values closer to 1. We first consider values of $x$ approaching 1 from the left ($x \lt 1$).

We now consider $x$ approaching 1 from the right ($x > 1$).

In both cases as $x$ approaches 1, $f(x)$ approaches 4. Intuitively, we say that $\lim_{{x \to 1}} f(x) = 4$.
NOTE: We are talking about the values that $f(x)$ takes when $x$ gets closer to $1$ and not $f(1)$. In fact, we may talk about the limit of $f(x)$ as $x$ approaches $a$ even when $f(a)$ is undefined.

Example 2

Let $g(x) = \dfrac{\sin x}{x}$ and compute $g(x)$ as $x$ takes values closer to 0. We consider values of $x$ approaching 0 from the left ($x \lt 0$) and values of $x$ approaching 0 from the right ($x > 0$).

Here we say that $\lim_{{x \to 0}} g(x) = 1$. Note that $g(0) = \dfrac{\sin 0}{0} = \dfrac{0}{0}$ is undefined at $x = 0$.

Graphical Approach to Limits

Example 3

The graph below shows that as $x$ approaches 1 from the left, $y = f(x)$ approaches 2 and this can be written as
$\lim_{{x \to 1^-}} f(x) = 2$
As $x$ approaches 1 from the right, $y = f(x)$ approaches 4 and this can be written as
$\lim_{{x \to 1^+}} f(x) = 4$
Note that the left and right hand limits and $f(1) = 3$ are all different.

Example 4

This graph shows that
$\lim_{{x \to 1^-}} f(x) = 2$
As $x$ approaches 1 from the right, $y = f(x)$ approaches 4 and this can be written as
$\lim_{{x \to 1^+}} f(x) = 4$
Note that the left hand limit $\lim_{{x \to 1^-}} f(x) = 2$ and $f(1) = 2$ are equal.

Example 5

This graph shows that
$\lim_{{x \to 0^-}} f(x) = 1$
and
$\lim_{{x \to 0^+}} f(x) = 1$
Note that the left and right hand limits are equal and we can write
$\lim_{{x \to 0}} f(x) = 1$
In this example, the limit when $x$ approaches 0 is equal to $f(0) = 1$.

Example 6

This graph shows that as $x$ approaches -2 from the left, $f(x)$ gets smaller and smaller without bound and there is no limit. We write
$\lim_{{x \to -2^-}} f(x) = -\infty$
As $x$ approaches -2 from the right, $f(x)$ gets larger and larger without bound and there is no limit. We write
$\lim_{{x \to -2^+}} f(x) = +\infty$
Note that $-\infty$ and $+\infty$ are symbols and not numbers. These are symbols used to indicate that the limit does not exist.

Example 7

The graph below shows a periodic function whose range is given by the interval [-1, 1]. If $x$ is allowed to increase without bound, $f(x)$ takes values within [-1, 1] and has no limit. This can be written
$\lim_{{x \to +\infty}} f(x) = \text{does not exist}$
If $x$ is allowed to decrease without bound, $f(x)$ takes values within [-1, 1] and has no limit again. This can be written
$\lim_{{x \to -\infty}} f(x) = \text{does not exist}$

Example 8

If $x$ is allowed to increase without bound, $f(x)$ in the graph below approaches 2. This can be written
$\lim_{{x \to +\infty}} f(x) = 2$
If $x$ is allowed to decrease without bound, $f(x)$ approaches 2. This can be written
$\lim_{{x \to -\infty}} f(x) = 2$

More References and links