The limit of f(x) as x approaches a is given by \lim_{x\to a}f(x)

Parts of the Limit Expression

limit command, with highlights for the parts described below.

Just as with logarithms and trigonometric functions, the basic command for a limit is a backslash followed by the letters you would write on paper.

\( \lim \)

When you enter this code into your project and compile the preview, the letters "lim" should appear in Roman font.

The expression that we are taking the limit of is simply typed after the limit command. For example, to take the limit of a function f(x), we would write

\( \lim f(x) \)

We also need to know what value the variable x is approaching. To express "x goes to a," we can type

\( x \to a \)

The command \to produces a thin arrow.

Finally, when writing a limit, the value that the variable approaches appears beneath the "lim." In order to get this effect, we use a subscript, indicated by an underscore, _. Remember to use curly braces to group the entire expression that belongs in the subscript.

\( \lim_{x \to a}f(x) \)

One-sided limits

For many functions, we also discuss the left-hand and right-hand limits.

Recall that the left-hand limit at a value is denoted by adding a - (minus sign). Similarly, the right-hand limit is denoted by adding a + (plus sign). Each is added as a superscript to the value in question. This superscript should be contained within the subscript of the limit.

Cartoon of left and right hand limits printed on hands.

Exercise 6.1.1

Typeset the following fact about limits:

If the limit as x approaches a from the right of f(x) equals the limit as x approaches a from the left of f(x) equals L, where L is a real number, then the limit as x approaches a of f(x) equals L.

Solution 6.1.1

The fact can be typeset with the following code:

If \(\lim_{x \to a^+}f(x) = \lim_{x \to a^-}f(x) = L\), where \(L\) is a real number, then \[\lim_{x \to a}f(x) = L.\]

Note that the + and - in the one-sided limits should be in superscript.

Symbol for infinity.


We also often encounter the idea of infinity when discussing limits.

The ∞ symbol for infinity can be obtained with the command \infty.

Exercise 6.1.2

Typeset the following limits involving infinity:

The limit as x approaches 0 from the right of 1 over x equals infinity
The limit as x approaches infinity of 1 over x is zero 0

Solution 6.1.2

The two limits can be typeset with the following code:

\[ \lim_{x\to\infty}\frac{1}{x} = 0 \]

\[ \lim_{x\to0^+}\frac{1}{x} = \infty \]

Note that both limits that were pictured are in display mode.

Display style for limits

Look back at the limits that you typeset in the exercises. They included both inline and display modes. What differences do you notice between the two?

In display mode, the subscript is contained entirely beneath the limit, for a more compact appearance. If you prefer this look, consider using the \displaystyle command when typesetting limits inline.

Limit in inline style:

Limit in inline style, described above.

Limit in display style:

Limit in display style, described above.