Limits and continuity

What a limit says

A limit describes the value a function approaches as its input gets close to a point, even when the function is not defined at that point. We write $\lim_{x\to a} f(x) = L$ . Numerically you evaluate $f$ at points closer and closer to $a$ , from both sides, and watch whether the values settle on one number.

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

(1)

A function is continuous at $a$ when the limit exists and equals $f(a)$ . There are three ways to fail: a hole (the limit exists but $f(a)$ is missing or different), a jump (the left and right limits differ) and a blow-up (no finite limit). An overnight gap in a price chart looks like a jump: the closing price on one side and the opening price on the other do not meet.

Limit: the value $f(x)$ settles on as $x$ approaches $a$ .
One-sided limit: the same, approaching only from the left ( $a^-$ ) or the right ( $a^+$ ).
Continuous at $a$ : the limit exists and equals $f(a)$ .
Removable hole: the limit exists but the function is undefined or different at the point.

A limit by numbersTake

f(x)=\frac{x^2-4}{x-2}

, undefined at

x=2

. At

x=1.9, 1.99, 1.999

it gives

3.9, 3.99, 3.999

; at

2.1, 2.01, 2.001

it gives

4.1, 4.01, 4.001

. Both sides head to 4, so the limit is 4. Factoring confirms it:

\frac{(x-2)(x+2)}{x-2}=x+2\to 4

Continuous compounding is a limit: $(1+\frac{1}{n})^n$ equals 2 at $n=1$ , 2.5937 at $n=10$ , 2.7048 at $n=100$ and 2.7169 at $n=1000$ , approaching $e\approx 2.71828$ .

Do not shrink the step foreverA computer keeps about 16 significant digits. If you try

h=10^{-20}

, then

x+h

is stored as

x

and the difference of two nearly equal numbers is pure rounding noise. Numerical limits and derivatives work best with a moderate step, around

10^{-5}

10^{-8}

Lab · what's graded

A limit with a hole15 pts
Compounding by numbers15 pts
A famous limit10 pts
Close to the limit15 pts
Kinds of discontinuity15 pts
Continuous at zero10 pts
A step that is too small20 pts

What a limit says

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

(1)

Limit: the value $f(x)$ settles on as $x$ approaches $a$ .
One-sided limit: the same, approaching only from the left ( $a^-$ ) or the right ( $a^+$ ).
Continuous at $a$ : the limit exists and equals $f(a)$ .
Removable hole: the limit exists but the function is undefined or different at the point.

A limit by numbersTake

f(x)=\frac{x^2-4}{x-2}

, undefined at

x=2

. At

x=1.9, 1.99, 1.999

it gives

3.9, 3.99, 3.999

; at

2.1, 2.01, 2.001

it gives

4.1, 4.01, 4.001

. Both sides head to 4, so the limit is 4. Factoring confirms it:

\frac{(x-2)(x+2)}{x-2}=x+2\to 4

Continuous compounding is a limit: $(1+\frac{1}{n})^n$ equals 2 at $n=1$ , 2.5937 at $n=10$ , 2.7048 at $n=100$ and 2.7169 at $n=1000$ , approaching $e\approx 2.71828$ .

Do not shrink the step foreverA computer keeps about 16 significant digits. If you try

h=10^{-20}

, then

x+h

is stored as

x

and the difference of two nearly equal numbers is pure rounding noise. Numerical limits and derivatives work best with a moderate step, around

10^{-5}

10^{-8}

Lab · what's graded

A limit with a hole15 pts
Compounding by numbers15 pts
A famous limit10 pts
Close to the limit15 pts
Kinds of discontinuity15 pts
Continuous at zero10 pts
A step that is too small20 pts

What a limit says

Lab · what's graded

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Limits and continuity

What a limit says

Lab · what's graded

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