MCV4U Limits: Graphs, Tables, Sequences and Continuity: Notes and Practice

In the last lesson the secant slopes for the falling stone crept toward $19.6$ but no secant ever had slope exactly $19.6$ ; setting $h = 0$ would mean dividing $0$ by $0$ . Mathematics needs a precise way to talk about the value something is heading toward, even when you are not allowed to land on it. That idea is the limit, and it is the engine of all of calculus.

x \to a lim f (x) = L

Read it as: the limit of f(x) as x approaches a is L. The values of f(x) can be made as close to L as you like by taking x close enough to a, from either side, but not equal to a.

The value of $f$ AT $a$ does not matter to the limit; only the values NEAR $a$ do. Take $f (x) = \frac{x ^{3} - 8}{x - 2}$ . At $x = 2$ it gives $\frac{0}{0}$ , which is undefined, but the table shows exactly what happens nearby.

x	1.9	1.99	1.999	2	2.001	2.01	2.1
$f (x)$	$11.41$	$11.9401$	$11.994001$	undefined	$12.006001$	$12.0601$	$12.61$

Values of (x³ − 8)/(x − 2) close to x = 2, from both sides.

From both sides the values close in on $12$ , so $lim_{x \to 2} \frac{x ^{3} - 8}{x - 2} = 12$ . Algebra confirms it. Factor the difference of cubes, $x^{3} - 8 = (x - 2) (x^{2} + 2 x + 4)$ , and cancel the common factor (allowed because $x \neq = 2$ while $x$ approaches $2$ ): the expression equals $x^{2} + 2 x + 4$ near $x = 2$ , which approaches $4 + 4 + 4 = 12$ . When direct substitution gives $\frac{0}{0}$ , simplify first, then substitute.

Reading limits from graphs, and one-sided limits

Fig 1Three ways a function can behave near x = 1: a hole in y = (x² − 1)/(x − 1), where the limit is 2 but f(1) is undefined; a jump, where the values approach 1.5 from the left but start at 3 on the right, so the limit does not exist; and a vertical asymptote of y = 1/(x − 1)², where the values grow without bound.

The jump in the middle graph shows why direction matters. A left-hand limit, written $lim_{x \to a^{-}} f (x)$ , uses only $x$ -values less than $a$ ; a right-hand limit, $lim_{x \to a^{+}} f (x)$ , uses only values greater than $a$ . The two-sided limit exists only when both one-sided limits exist and are equal. In the jump graph the left-hand limit is $1.5$ and the right-hand limit is $3$ , so $lim_{x \to 1} f (x)$ does not exist (DNE). In the asymptote graph the values grow without bound, written $lim_{x \to 1} f (x) = \infty$ ; that describes the behaviour, but it is not a finite limit.

Limits also describe what happens far out along a graph. As $x$ grows, $g (x) = \frac{2 x + 1}{x - 3}$ gets closer and closer to its horizontal asymptote $y = 2$ , written $lim_{x \to \infty} g (x) = 2$ . Near its vertical asymptote $x = 3$ , it shoots up on the right and down on the left.

x	10	100	1000	10 000
$\frac{2 x + 1}{x - 3}$	$3$	$2.072165$	$2.007021$	$2.000700$

As x grows, the values approach the horizontal asymptote y = 2 without ever reaching it. (Near x = 3: 72 at x = 3.1 and −68 at x = 2.9.)

VideoJensenMath, an Ontario teacher, reads left-hand and right-hand limits off piecewise graphs and checks whether the two-sided limit exists, as in this section.Watch on YouTube

Limits of sequences

Fig 2Left: the terms of (1 + 1/n)ⁿ for n = 1 to 12 rise toward the dashed line y = e ≈ 2.718. Right: the ratios of consecutive Fibonacci numbers alternate above and below the dashed line y = φ ≈ 1.618 and close in on it.

n	(1 + 1/n) to the power n	Fibonacci ratio	Partial sum of 4(1 − 1/3 + 1/5 − …), n terms
$1$	$2$	$1$	$4$
$2$	$2.25$	$2$	$2.666667$
$3$	$2.370370$	$1.5$	$3.466667$
$4$	$2.441406$	$1.666667$	$2.895238$
$5$	$2.488320$	$1.6$	$3.339683$
$10$	$2.593742$	$1.618182$	$3.041840$
$100$	$2.704814$	$1.618034$	$3.131593$
$1000$	$2.716924$	$1.618034$	$3.140593$
Limit	$e \approx 2.718282$	$φ \approx 1.618034$	$π \approx 3.141593$

Three famous sequences, rounded to 6 decimal places. The Fibonacci ratio is the next Fibonacci number divided by the current one (1, 1, 2, 3, 5, 8, …).

A sequence has a limit $L$ if its terms get as close to $L$ as you like and stay that close once $n$ is large enough, written $lim_{n \to \infty} t_{n} = L$ . The three sequences approach in different styles. The terms $(1 + \frac{1}{n})^{n}$ climb steadily toward the number $e$ , which returns later in this course as the base of the exponential function. The Fibonacci ratios alternate above and below the golden ratio $φ = \frac{1 + 5}{2}$ and settle down quickly. The sums $4 (1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots)$ also alternate, but they creep toward $π$ very slowly: after 1000 terms they are still off in the third decimal place.

Continuity

A function is continuous at $x = a$ when three things hold: $f (a)$ is defined, $lim_{x \to a} f (x)$ exists, and the two are equal. Informally, you can draw the graph through $x = a$ without lifting your pencil.
A removable discontinuity (a hole): the limit exists, but $f (a)$ is undefined or has a different value.
A jump discontinuity: the left-hand and right-hand limits exist but are different.
An infinite discontinuity: the function grows without bound near a vertical asymptote.
Polynomials are continuous everywhere, so for a polynomial $lim_{x \to a} f (x) = f (a)$ : just substitute.

Limits: Graphs, Tables, Sequences and Continuity

Key things to remember

Reading limits from graphs, and one-sided limits

Limits of sequences

Continuity

Worked examples

Where marks get lost

Flashcards 8

Practice