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Limits of Functions

Definition

A limit is the (output) value that a function or sequence approaches as the input approaches some particular value. The notation for limits is

\[\displaystyle{\lim_{x\rightarrow c} f(x) = l}\]

which means: the limit of the function $f(x)$ as $x$ tends to $c$ is $l$.

One Sided Limits

Alternatively, $x$ may approach $c$ from above (right) or below (left), in which case the limits may be written as: \[\displaystyle{\lim_{x\to c^+} f(x) = l} \;\text { or }\;\displaystyle{\lim_{x\to c^-} f(x) = l}\]

respectively.

If both of these limits are equal to $l$, then this can be referred to as the limit of $f(x)$ at $c$. Conversely, if they are both not equal to $l$, then the limit does not exist.

Properties

Properties of limits as $x$ approaches some value $c$.

  • $\displaystyle{\lim_{x\to c} \bigl(f(x)+g(x)\bigr) = \lim_{x\to c} f(x) +\lim_{x\to c} g(x)}$
  • $\displaystyle{\lim_{x\to c} \bigl(f(x)-g(x)\bigr) = \lim_{x\to c} f(x) -\lim_{x\to c} g(x)}$
  • $\displaystyle{\lim_{x\to c} \bigl(f(x) \times g(x)\bigr) = \lim_{x\to c} f(x) \times \lim_{x\to c} g(x)}$
  • $\displaystyle{\lim_{x\to c} \left(\frac{f(x)}{g(x)}\right) = \frac{\lim_{x\to c} f(x)}{\lim_{x\to c} g(x)}}$

As $x\rightarrow\infty$

The function $f(x) \rightarrow l$, a real limit, as $x\rightarrow\infty$ if, however small a distance we choose, $f(x)$ gets closer than this distance to $l$.

The function $f(x)\rightarrow \infty$ as $x\rightarrow\infty$ if, however large a number we choose, $f(x)$ gets larger than this number.

The function $f(x) \rightarrow -\infty$ as $x\rightarrow\infty$ if, however large and negative a number we choose, $f(x)$ gets more negative than this number.

As $x\rightarrow -\infty$

The function $f(x) \rightarrow l$, a real limit, as $x\rightarrow -\infty$ if, however small a distance we choose, $f(x)$ gets closer than this distance to $l$.

The function $f(x)\rightarrow \infty$ as $x\rightarrow -\infty$ if, however large a number we choose, $f(x)$ gets larger than this number.

The function $f(x) \rightarrow -\infty$ as $x\rightarrow -\infty$ if, however large and negative a number we choose, $f(x)$ gets more negative than this number.

As $x\rightarrow c$

The limit of $f(x)$ as $x$ tends to a real number $c$, is the value $f(x)$ approaches as $x$ gets closer to that real number.

Common Limits

Whilst it is necessary to know how to work out all limits, it is useful to know some that appear often.

  • $\displaystyle{\lim_{x\to c} a = a}$
  • $\displaystyle{\lim_{x\to c} x^r = c^r}$ for $r \geq 1$
  • $\displaystyle{\lim_{x\to 0^+} \frac{1}{x^r} = \infty}$
  • $\displaystyle{\lim_{x\to 0^-} \frac{1}{x^r} =

\begin{cases} -\infty, \;\;\text{if } r \text{ is odd}\\ +\infty, \;\;\text{if } r \text{ is even}

\end{cases}}$

  • $\displaystyle{\lim_{x\to \infty} x^N =

\begin{cases} \begin{align} \infty, \;\;&N\gt 0\\ 1, \;\; &N=0\\ 0, \;\;&N\lt 0 \end{align} \end{cases}}$

  • $\displaystyle{\lim_{x\to \infty} N^x =

\begin{cases} \begin{align} \infty, \;\;&N\gt 1\\ 1, \;\; &N=1\\ 0, \;\;&0\lt N \lt 1. \end{align} \end{cases}}$

  • $\displaystyle{\lim_{x\to \infty} \frac{x}{N} =

\begin{cases} \begin{align} \infty, \;\;&N\gt 0\\ \text{does not exist}, \;\; &N=0\\ -\infty, \;\;&N\lt 0 \end{align} \end{cases}}$

  • $\displaystyle{\lim_{x\to \infty} \frac{N}{x} = 0}$ for any real $N$

Worked Examples

Example 1

Find the limit of the function $f(x) = \dfrac{1}{x}$ as $x\rightarrow\infty$.

Solution

To start, put a few increasing values of $x$ into the function to get an idea of the behaviour of $f(x)$. It may also be helpful to look at the graph of $f(x) = \dfrac{1}{x}$.

$x$

1

2

3

4

10

20

100

$f(x)$

1

$\dfrac{1}{2}$

$\dfrac{1}{3}$

$\dfrac{1}{4}$

$\dfrac{1}{10}$

$\dfrac{1}{20}$

$\dfrac{1}{100}$

We can see that as the values of $x$ increase, the value of $f(x)$ is getting closer to zero, and we know the function $\dfrac{1}{x}$ can never be equal to zero.

So we say $f(x) \rightarrow 0$ as $x\rightarrow\infty$ or $\displaystyle{\lim_{x\rightarrow\infty} \frac{1}{x}~} = 0$.

Example 2

Find the limit of the function $f(x) = x^3$ as $x\rightarrow -\infty$.

Solution

To start, put a few decreasing values of $x$ into the function to get an idea of the behaviour of $f(x)$. It may also be helpful to look at the graph of $f(x) = x^3$.

$x$

0

-1

-2

-3

-10

-20

-100

$f(x)$

0

-1

-8

-27

-1000

-8000

-1000000

We can see that as the values of $x$ decrease, the value of $f(x)$ is clearly getting more and more negative.

So we say $f(x) \rightarrow -\infty$ as $x\rightarrow -\infty$ or $\displaystyle{\lim_{x\rightarrow -\infty} x^3} = -\infty$.

Example 3

Find the limit of the function $f(x) = x+4$ as $x\rightarrow 1$.

Solution

As the function exists at $x=1$, we can just put this value into the function and find a value for $f(x)$.

So we have that $f(x) \rightarrow 5$ as $x\rightarrow 1$ or $\displaystyle{\lim_{x\rightarrow 1} (x+4)} = 5$.

Video Examples

Example 1

Prof. Robin Johnson finds the limit of $\dfrac{x^2+1}{x^3+1}$ as $x\rightarrow 1$.

Example 2

Prof. Robin Johnson finds the limit of $\dfrac{x^2-4}{x-2}$ as $x\rightarrow 2$.

Example 3

Prof. Robin Johnson finds the limit of $\dfrac{\sin 3x}{\tan x}$ as $x\rightarrow 0$.

External Resources

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