# Introduction to Limits in Calculus

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

## 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 $$x = 0$$.

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

Calculus Tutorials and Problems
Limits of Absolute Value Functions Questions
What is the limit of a function?