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

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.

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 \( x = 0 \).

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.

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.

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 \( f(0) = 1 \).

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.

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

\( \lim_{{x \to -\infty}} f(x) = \text{does not exist} \)

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

\( \lim_{{x \to -\infty}} f(x) = 2 \)

Limits of Absolute Value Functions Questions

What is the limit of a function?