IB Analysis and Approaches HL · Functions · Section 9 of 15

Limit of a function: intuitive idea

3 worked examples · By Álvaro López

Key idea

A limit describes the value that $f (x)$ approaches as $x$ gets close to a number, regardless of what happens at that number. It also tells you what the function does when $x$ grows or decreases without bound. This is how asymptotes are read from a graph, and it is the basis of continuity and the derivative. In the IB exam it appears mainly when reading graphs and tables.

Approaching a point

Imagine you trace the graph with your finger towards the point with $x$ -coordinate $a$ and stop just before you reach it. Look at the height your finger is at. You can arrive from the left ( $x < a$ ) or from the right ( $x > a$ ), and you do not always arrive at the same height.

Function with a jump at x equal to a: from the left it approaches L (open point), from the right L' (open point) and the value f(a) is a closed point in between

In the diagram, from the left the function approaches $L = 2$ and from the right $L^{'} = 4$ . The filled point $f (a) = 3$ matches neither, and that does not matter: the limit is about where the function is heading, not where it is.

The value of the limit does not depend on the value of $f (a)$ , or even on whether $f (a)$ exists. You can have a limit at a point where the function is not defined, such as a "hole" in the graph.

At an endpoint of the domain only the side that exists makes sense: if the domain starts at $- 3$ , you can only calculate $x \to - 3^{+} lim f (x)$ .

When the values blow up

Sometimes, as $x$ approaches $a$ , the values of $f (x)$ grow without bound or decrease without bound. This is written $x \to a^{+} lim f (x) = + \infty$ or $- \infty$ , depending on the case. The symbols $+ \infty$ and $- \infty$ are not numbers: they describe the behaviour of the function, not a value that is reached.

If $f$ is a quotient, the candidates for vertical asymptotes are the values of $x$ that make the denominator zero. They are only asymptotes if the limit genuinely blows up; some values that make the denominator zero give a hole, not an asymptote (see the next section).

When $x$ becomes very large

You can also study the behaviour "far away", with $x \to + \infty$ (large positive values) or $x \to - \infty$ (large negative values). There are three possibilities:

the values of $f (x)$ approach a number $L$ : $x \to + \infty lim f (x) = L$ ;
they grow without bound: $x \to + \infty lim f (x) = + \infty$ ;
they decrease without bound: $x \to + \infty lim f (x) = - \infty$ .

The same holds for $x \to - \infty$ .

Graph of y = 1/(x-1) + 2 with the vertical asymptote x = 1 and the horizontal asymptote y = 2, both dashed

Here $x \to 1^{-} lim f (x) = - \infty$ , $x \to 1^{+} lim f (x) = + \infty$ and $x \to \pm \infty lim f (x) = 2$ . From this you can read off directly $x = 1$ as the vertical asymptote and $y = 2$ as the horizontal asymptote.

Worked examples

Example 1 Basic

Limits of a piecewise-defined function

A function is defined as follows:

$f (x) = ⎩ ⎨ ⎧ x + 2 1 6 - x if x < 1 if x = 1 if x > 1$

a) Calculate $x \to 1^{-} lim f (x)$ .

b) Calculate $x \to 1^{+} lim f (x)$ .

c) Does $x \to 1 lim f (x)$ exist? Answer "yes" or "no".

d) Calculate $x \to 4 lim f (x)$ .

Solution

a) For $x < 1$ the rule $x + 2$ is used. As $x$ approaches $1$ from the left, $x + 2$ approaches $3$ . Therefore $x \to 1^{-} lim f (x) = 3$ .

b) For $x > 1$ the rule $6 - x$ is used, which approaches $5$ . Therefore $x \to 1^{+} lim f (x) = 5$ .

c) The one-sided limits are different ( $3 \neq = 5$ ), so the limit at $x = 1$ does not exist. The value $f (1) = 1$ changes nothing.

d) Near $x = 4$ the rule $6 - x$ is used on both sides, and it approaches $2$ : $x \to 4 lim f (x) = 2$ .

Example 2 Medium

Asymptotes of a quotient

Let $f (x) = \frac{3 x + 1}{x - 2}$ .

a) Calculate $x \to 2^{+} lim f (x)$ and $x \to 2^{-} lim f (x)$ .

b) Calculate $x \to + \infty lim f (x)$ .

c) Write down the equations of the asymptotes.

Solution

a) The denominator is zero at $x = 2$ , and the numerator equals $3 \cdot 2 + 1 = 7 > 0$ near that point.

With $x$ slightly greater than $2$ , the denominator is a very small positive number, so the quotient is positive and huge: $x \to 2^{+} lim f (x) = + \infty$ .
With $x$ slightly less than $2$ , the denominator is a very small negative number: $x \to 2^{-} lim f (x) = - \infty$ .

A table confirms the sign: $f (2.01) = \frac{7.03}{0.01} = 703$ and $f (1.99) = \frac{6.97}{- 0.01} = - 697$ .

b) When $x$ is very large, the $+ 1$ and the $- 2$ count for very little compared with $3 x$ and $x$ , so $f (x) \approx \frac{3 x}{x} = 3$ . A table confirms it: $f (1000) = \frac{3001}{998} \approx 3.007$ . Therefore $x \to + \infty lim f (x) = 3$ .

c) The vertical asymptote is $x = 2$ and the horizontal asymptote is $y = 3$ .

Example 3 Hard

Parameters from the asymptotes

The function $f (x) = \frac{a x + 3}{x - b}$ , with $a$ and $b$ constants, has the line $x = 2$ as its vertical asymptote and the line $y = - 3$ as its horizontal asymptote.

a) Find the values of $a$ and $b$ .

b) With those values, calculate $x \to 2^{-} lim f (x)$ and $x \to 2^{+} lim f (x)$ .

c) Find the point at which the graph cuts the $y$ -axis.

Solution

a) A vertical asymptote is where the denominator is zero: $x - b = 0$ gives $x = b$ , and since the asymptote is $x = 2$ , we get $b = 2$ .

For the horizontal one, with $x$ very large, $f (x) \approx \frac{a x}{x} = a$ . The asymptote is $y = - 3$ , so $a = - 3$ .

b) Now $f (x) = \frac{- 3 x + 3}{x - 2}$ . Near $x = 2$ the numerator is $- 3 \cdot 2 + 3 = - 3 < 0$ .

If $x \to 2^{-}$ , the denominator is negative and small, and the quotient is (negative)/(negative small) = positive and huge: $x \to 2^{-} lim f (x) = + \infty$ .
If $x \to 2^{+}$ , the denominator is positive and small: $x \to 2^{+} lim f (x) = - \infty$ .

c) With $x = 0$ : $f (0) = \frac{3}{- 2} = - \frac{3}{2}$ . The point is $(0, - \frac{3}{2})$ .

In the exam: when giving the asymptotes, write their equations ( $x = 2$ , $y = - 3$ ), not just the numbers. In a limit question with $\pm \infty$ , reason about the sign of the numerator and of the denominator: without that reasoning you lose half the marks.

Common mistakes

Summary

$x \to a^{-} lim f (x)$ and $x \to a^{+} lim f (x)$ are the one-sided limits. The limit $x \to a lim f (x)$ exists only if they coincide.
The limit does not depend on $f (a)$ (which may not exist).
$x \to a lim f (x) = \pm \infty$ from one side gives a vertical asymptote $x = a$ ; in a quotient, look for where the denominator is zero.
$x \to \pm \infty lim f (x) = L$ gives a horizontal asymptote $y = L$ .
Reading a graph: trace the curve from each side; in Paper 2, a table of values or the GDC graph serves as a check.

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Limit of a function: intuitive idea

Approaching a point

When the values blow up

When x becomes very large

Worked examples

Common mistakes

Summary

Want to work on it together?

When $x$ becomes very large