# Introduction to Limits in Calculus

We intend to give a numerical and graphical approaches to the concept of limits using examples.

## Numerical Approach to Limits

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

 $x$ $f(x)$ 0.5 3 0.8 3.6 0.9 3.8 0.95 3.9 0.99 3.98 0.999 3.998 0.9999 3.9998 0.99999 3.99998

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

 $x$ $f(x)$ 1.5 5 1.2 4.4 1.1 4.2 1.05 4.1 1.01 4.02 1.001 4.002 1.0001 4.0002 1.00001 4.00002

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 $(x < 0)$ and values of $x$ approaching $0$ from the right $(x > 0)$.

 $x$ $g(x)$ -0.5 0.9588 -0.2 0.993346 -0.1 0.998334 -0.01 0.999983 -0.001 0.999999

 x g(x) 0.5 0.9588 0.2 0.993346 0.1 0.998334 0.01 0.999983 0.001 0.999999

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

## 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 $f(1)$ have different values.

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 and $f(1)$ have equal values.

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 $f(0)$.

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

$\lim_{x \to +\infty} f(x)$ does not exist

If $x$ is allowed to decrease without bound, $f(x)$ take values within $[-1 , 1]$ and has no limit again. This can be written

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

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

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Updated: 2 April 2013