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 (\( x \lt 1 \)).
Table of Values of f(x) as x Approaches 1 From Left

Table of Values of f(x) as x Approaches 1 From Left

We now consider \( x \) approaching 1 from the right (\( x > 1 \)).
Table of Values of f(x) as x Approaches 1 From Right

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 (\( x > 0 \)).

Table of Values of g(x) as x Approaches 0 From the Left and from the Right

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 \[ \lim_{{x \to -\infty}} f(x) = \text{does not exist} \] graph example 7

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