Limits, via floating point

Quotients and limits

Question 1

What is $\lim_{x \to 0} \dfrac{|x|}{x}$ ?

A.0, since

|x|

and

x

both go to 0B.It does not exist: the two sides head for

-1

and

1

C.1, since

|x|

and

x

have the same sizeD.It is

1

or

-1

, and which one depends on the value of

x

you choose

Question 2

What does $\lim_{h \to 0} q(h) = 6$ assert?

A.That

q(0) = 6

, since setting

h

to zero is what the arrow meansB.For every tolerance you name, all

h

close enough to 0, but not equal to 0, keep

q(h)

within that tolerance of 6C.That

q(h)

eventually reaches 6 and then stops changingD.That

q(h)

equals 6 once

h

is small enough to be stored as zero

Question 3

What is the difference quotient of $f(x) = x^2$ at $a = 2$ with $h = 0.1$ ?

A.0.41, the difference between the two function valuesB.4.1C.4, because

h

is small enough to ignoreD.2.1, the second of the two points

Question 4

$g(x) = \dfrac{x^2 - 9}{x - 3}$ . What is $\lim_{x \to 3} g(x)$ ?

A.0, because the top of the fraction is 0 when

x = 3

B.Infinite, because the bottom of the fraction is 0 when

x = 3

C.It does not exist, because

g(3)

is undefinedD.6

Floating point and continuity

Question 5

$g(x) = \dfrac{x^2 - 9}{x - 3}$ for $x \neq 3$ , and $g(3)$ is defined to be 5. Which is true?

A.

g

is continuous at 3, because

g(3)

now existsB.The limit at 3 is now 5, because a limit takes the value at the pointC.The limit at 3 is 6 but

g(3)

is 5, so

g

is not continuous at 3D.The limit does not exist, because the formula still divides by zero at 3

Question 6

You want the best numerical estimate of the slope of $f$ at $a$ from a difference quotient in doubles. Which step size should you use?

A.Exactly 0, since that is what the limit saysB.Around

10^{-16}

, matching the precision of a doubleC.As small as the language will let you write, since smaller

h

is always closer to the limitD.Around

10^{-8}

, which gives roughly eight correct significant digits

Question 7

A car park is free for the first 15 minutes and costs 1.50 after that. Taking the time in minutes as $m$ , what is $\lim_{m \to 15^+} \text{charge}(m)$ , and what is $\text{charge}(15)$ ?

A.Both are 0, because 15 minutes is still inside the free periodB.Both are 1.50, because the limit is what the function is about to becomeC.The limit from the right is 1.50 and the value at 15 is 0, so the charge is not continuous at 15D.The limit does not exist, because a charge cannot be defined between whole minutes

Question 8

In a program using doubles, is 3.0 + 1e-17 == 3.0 true or false?

A.True: the sum needs more significant digits than a double holds, so it rounds back to 3.0B.True, but only in some languages; others keep the extra digitsC.False: two different numbers were added, so the result must differ from 3.0D.False: 1e-17 is too small to store, so the addition raises an error

Question 9

Computing $\frac{f(a+h) - f(a)}{h}$ in doubles, the answers stop improving below about $h = 10^{-8}$ . Why?

A.Dividing by a very small number overflows, and the result is clampedB.The mathematical error term grows again once

h

is small enoughC.Below

10^{-8}

the language quietly switches to a lower precision typeD.Subtracting two nearly equal doubles cancels the digits they share, leaving a short and mostly rounded result, and dividing by a tiny

h

magnifies that error

9 questions left to answer.