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

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

• Limits of functions workbook at mathcentre.