The Concept Of A Limit

In this section, we introduce the concept of a limit intuitively. We examine how a function behaves near a given input value using numerical values, graphical illustrations, and algebraic simplification.

Quick Reference

Concept	Symbolic Notation	Description
Intuitive Limit	$lim_{x \to a} f (x) = L$	As x approaches a (with $x \neq a$ ), $f (x)$ gets close to $L$ .
Dynamic Notation	$f (x) \to L$ as $x \to a$	Alternative notation for describing the limiting behavior.
Uniqueness Theorem	$lim_{x \to a} f (x) = L_{1}, L_{2} ⟹ L_{1} = L_{2}$	If a limit exists, its value is unique.
Loss of Significance	Numerical Rounding Error	Subtraction of nearly equal numbers in finite-precision computing can generate false limits.

Consider the function f defined by the equation

f (x) = \frac{2 x^{2} - 2}{x - 1}

The function f is defined for all values of x except $x = 1$ because the substitution of $x = 1$ in the expression for $f (x)$ yields the undefined fraction $\frac{0}{0}$ . But because $2 x^{2} - 2 = 2 (x^{2} - 1) = 2 (x - 1) (x + 1)$ , if $x \neq 1$ , we can simplify the fraction as

f (x) = \frac{2 x^{2} - 2}{x - 1} = \frac{2 (x - 1) (x + 1)}{x - 1} = 2 x + 2 (x \neq 1)

So the graph of f is the line $y = 2 x + 2$ with one point removed, namely $(1, 4)$ . This point is shown as a hole in the following figure.

TikZ figure — Graph of $f (x) = \frac{2 x^{2} - 2}{x - 1}$ .

Now let's investigate the values of f when x is close to 1 but not equal to 1. Let x take on the values $0.9, 0.95, 0.99, 0.999, 0.9999$ , and so on, or take on the values $1.1, 1.05, 1.01, 1.001, 1.0001$ , and so on. The corresponding values of f are shown in the following table.

Table of values for f(x) as x approaches 1 from left and right — Values of $f (x)$ for x near $1$ .

From this table and graph of f, we see that as x gets closer and closer to 1 (on either side of 1), but not equal to 1, $f (x)$ gets closer and closer to 4; the closer x is to 1, the closer $f (x)$ is to 4. More specifically, we can make the values of $f (x)$ as close to 4 as we desire by taking x close enough to 1. We express this by saying that "the limit of $f (x)$ as x approaches 1 is 4" or simply " $f (x)$ approaches 4 as x approaches 1," and express it symbolically as

lim_{x \to 1} f (x) = 4

f (x) \to 4 as x \to 1

In general:

Intuitive Definition of a Limit: If we can make the values of $f (x)$ as close as we please to a number $L$ by taking x sufficiently close (but not equal) to a, we say "the limit of $f (x)$ as x approaches a is $L$ " and write

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

f (x) \to L as x \to a .

According to the above definition, x approaches a but $x \neq a$ , so the non-existence or existence of $f (a)$ or its value (if it exists) has no bearing on the existence or on the value of $lim_{x \to a} f (x)$ . For example, if we define the function g as

g(x) = \begin{cases} \dfrac{2x^2 - 2}{x - 1} & \text{if } x \neq 1 \\[6pt] 3 & \text{if } x = 1 \end{cases}

then $g (x)$ and $f (x) = \frac{2 x^{2} - 2}{x - 1}$ are basically the same except when $x = 1$ ; therefore, because $lim_{x \to 1} f (x) = 4$ , we have

lim_{x \to 1} g (x) = 4.

Numerical and Graphical Approaches

For now, to evaluate the limits, we use numerical and graphical approaches.

Evaluate $lim_{x \to - 1.2} x^{2}$ numerically.

Solution

Let x approach $- 1.2$ from both sides:

Table of values for y = x^2 near x = -1.2 — Values of $y = x^{2}$ for x near $- 1.2$ .

As the above table shows and as we expect from the graph of $y = x^{2}$ (see the following figure), as x approaches $- 1.2$ , $y$ approaches $(- 1.2)^{2} = 1.44$ .

Evaluate $lim_{x \to 2} \frac{x^{2} - 5 x + 6}{x - 2}$ numerically.

Solution

Let's choose some values of x close to 2 and calculate the corresponding values of $y = (x^{2} - 5 x + 6) / (x - 2)$ . The results are tabulated in the following table.

Table of values for (x^2 - 5x + 6)/(x - 2) near x = 2 — Values near $x = 2$ .

It appears as if $y$ were approaching the limit $- 1$ . To verify this, we factor the numerator and simplify the fraction as

y = \frac{x^{2} - 5 x + 6}{x - 2} = \frac{(x - 2) (x - 3)}{x - 2} = x - 3 (if x \neq 2)

From this, it appears that $y$ can be made as near $- 1$ as we please by taking x sufficiently close to 2 (see the following figure). Hence $- 1$ is the limit of $y$ as x approaches 2.

Graph of y = (x^2 - 5x + 6)/(x - 2) — Graph of $y = \frac{x^{2} - 5 x + 6}{x - 2}$ .

Evaluate $lim_{x \to 0} \frac{\sin x}{x}$ numerically.

Solution

Again we note that the function $y = \sin x / x$ is not defined at $x = 0$ , but we can construct the following table by choosing some values of x close to 0 and calculating the corresponding values of $\sin x / x$ . Recall that $\sin x$ means the angle x is measured in radians, so you need to set your calculator in radian mode.

Table of values for sin(x)/x near x = 0 — Values of $\frac{\sin x}{x}$ for x close to $0$ .

The graph of $y = \sin x / x$ is shown below. From the above table and this figure, we may conclude that

lim_{x \to 0} \frac{\sin x}{x} = 1.

When the Numerical Approach Fails

In the previous examples, we used a calculator/computer to numerically evaluate the values of the given function $f (x)$ for x near the given point. However, in some cases, computers may give false results. Here is an example of such a situation.

Use the numerical approach to evaluate $lim_{x \to 0} \frac{1 - \sqrt{1 + x^{6}}}{x^{6}}$ .

Solution

The function

f (x) = \frac{1 - \sqrt{1 + x^{6}}}{x^{6}}

is not defined when $x = 0$ . Let's construct a table to list the values of $f (x)$ for several values of x near 0.

Table 5 showing f(x) values near x = 0 — Values of $f (x)$ near $x = 0$ .

From this table, we may conclude that

lim_{x \to 0} \frac{1 - \sqrt{1 + x^{6}}}{x^{6}} = - 0.5

We might be tempted to choose some values of x closer to 0. Let's try it out.

Table 6 showing breakdown of calculation near 0 — Values of $f (x)$ for x extremely close to $0$ .

The values in the above table are given by WolframAlpha.com. If you use a calculator, you might get different values, but eventually, you will get 0 if you make x sufficiently close to zero. Do these calculations show us the limit is 0 instead of $- 0.5$ ? Let's graph this function. Again it seems that the limit is $- 0.5$ .

Graph of f(x) when -3 <= x <= 3 — Graph of $f (x) = \frac{1 - \sqrt{1 + x^{6}}}{x^{6}}$ when $- 3 \leq x \leq 3$ .

Let's zoom in and graph this function again for values of x closer to 0.

Graph of f(x) when -0.2 <= x <= 0.2 — Graph of $f (x) = \frac{1 - \sqrt{1 + x^{6}}}{x^{6}}$ when $- 0.2 \leq x \leq 0.2$ .

What is the reason for this strange behavior? This behavior originates from the fact that calculators and computers retain only a fixed number of digits during a calculation. For example, if a calculator stores only 4 significant digits, then 0.34246 and 0.34254 are both stored as 0.3425. In this specific example, when x is very close to 0, $\sqrt{1 + x^{6}}$ is so close to 1 that the computer cannot distinguish between them. This undesirable effect is called loss of significance. It almost always occurs when subtracting two nearly equal numbers. Ways to avoid loss of significance are studied in numerical methods.

Here, we can multiply both the numerator and the denominator by the conjugate of the numerator:

\begin{aligned} \frac{1 - \sqrt{1 + x^6}}{x^6} &= \frac{1 - \sqrt{1 + x^6}}{x^6} \frac{1 + \sqrt{1 + x^6}}{1 + \sqrt{1 + x^6}} \\ &= \frac{1 - \left(\sqrt{1 + x^6}\right)^2}{x^6 \left(1 + \sqrt{1 + x^6}\right)} \\ &= \frac{1 - (1 + x^6)}{x^6 \left(1 + \sqrt{1 + x^6}\right)} \\ &= \frac{-x^6}{x^6 (1 + \sqrt{1 + x^6})} \\ &= -\frac{1}{1 + \sqrt{1 + x^6}} \end{aligned}

Note that

y = \frac{1 - \sqrt{1 + x^{6}}}{x^{6}} and y = - \frac{1}{1 + \sqrt{1 + x^{6}}}

are equal except when $x = 0$ because the first one is not defined for $x = 0$ but the second one is. The following table lists the values of $y = - \frac{1}{1 + \sqrt{1 + x^{6}}}$ for several points approaching $x = 0$ .

Table 7 showing corrected calculations — Values of $y = - \frac{1}{1 + \sqrt{1 + x^{6}}}$ near $x = 0$ .

Calculations with WolframAlpha.com and the graph of $y = - \frac{1}{1 + \sqrt{1 + x^{6}}}$ are shown below.

Graph of y = -1/(1 + sqrt(1 + x^6)) — Calculations with WolframAlpha and graph of $y = - \frac{1}{1 + \sqrt{1 + x^{6}}}$ .

Now let's consider the sign function $y = sgn (x)$ .

Let the sign function be defined by

\operatorname{sgn}(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases}.

Find $lim_{x \to 0} sgn (x)$ .

Solution

The graph of this function is shown below. When x approaches 0 through positive numbers, $sgn (x)$ approaches 1. Although $sgn (x) = 1$ for every $x > 0$ , we still say $sgn (x)$ approaches 1 through positive numbers (or from the right).

When x approaches 0 through negative numbers, $sgn (x)$ approaches $- 1$ . Therefore, there is no single number that $sgn (x)$ approaches as x approaches 0 from either side. Therefore, $lim_{x \to 0} sgn (x)$ does not exist.

In general, the limit of $f (x)$ as x approaches a number a (if it exists) is unique, because for all x near a, $f (x)$ cannot be near two different numbers at the same time.

Uniqueness of a Limit Theorem: If $lim_{x \to a} f (x) = L_{1}$ and $lim_{x \to a} f (x) = L_{2}$ , then $L_{1} = L_{2}$ .

The precise definition of this theorem once the precise definition of a limit is explained is not difficult.

Quick Reference

Numerical and Graphical Approaches

When the Numerical Approach Fails

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