Unit 1 check

Unit 1 check: Limits and Continuity

15 questions from 15 lessons in Unit 1. Answer each one, and if you miss, follow the link back to where it is taught, then try again. Your score counts first tries.

Question 1 · from 1.1 · Calculate

No calculator. A phone is plugged in to charge. Its battery level is $B (t)$ percent, $t$ minutes after it is plugged in, with $B (0) = 15$ , $B (20) = 44$ and $B (50) = 77$ . What is the average rate of change of $B$ , in percent per minute, from $t = 20$ to $t = 50$ ?

Your answer

Question 2 · from 1.2 · Match

No calculator. Tap a statement in notation, then the sentence that says the same thing.

Question 3 · from 1.3 · Sort it

No calculator. None of these limits exists. Tap a card, then tap the reason it fails.

The two sides disagree

The function is unbounded

The function oscillates

Question 4 · from 1.4 · Decide and justify

No calculator. A table gives only three values: $p (0.1) = 3.31$ , $p (0.01) = 3.0301$ and $p (0.001) = 3.003001$ . A classmate says the table shows that $\lim_{x \to 0} p (x)$ is about 3. Is the classmate right?

Which reason supports your answer?

Question 5 · from 1.5 · Decide and justify

No calculator. Suppose $\lim_{x \to 0} f (x) = 0$ and $\lim_{x \to 0} g (x) = 0$ . A classmate argues that $\lim_{x \to 0} \frac{f (x)}{g (x)}$ does not exist, because the quotient theorem would give $\frac{0}{0}$ . Does the quotient theorem show that the limit does not exist?

Which reason supports your answer?

Setup for question 6

A function $g$ is defined by one formula for every $x$ except 3 and a separate value at 3, as the table shows.

The definition of g
Input	$g (x)$
$x \neq 3$	$\frac{x^{2} - x - 6}{x - 3}$
$x = 3$	$1$

Question 6 · from 1.6 · Calculate

No calculator. Find $\lim_{x \to 3} g (x)$ .

Your answer

Setup for question 7

A function $g$ is defined for all real numbers. Its formula is not given, but $g (2) = 5$ , and the table shows $g (x)$ at inputs close to 2.

Values of g near x = 2
$x$	$g (x)$
1.9	11.41
1.99	11.9401
1.999	11.994001
2.001	12.006001
2.01	12.0601
2.1	12.61

Question 7 · from 1.7 · Multiple choice

No calculator. Which statement about $g$ near $x = 2$ do the table and the value $g (2) = 5$ support?

Question 8 · from 1.8 · Spot the error

No calculator. A student uses the squeeze theorem to find $\lim_{x \to 0} x^{3} \cos (\frac{2}{x})$ . Tap the first line with a mistake.

Question 9 · from 1.9 · Sort it

No calculator. Each card describes a function $g$ near $x = 3$ . Tap a card, then the bin for what it shows about $\lim_{x \to 3} g (x)$ .

The limit is 2

The limit does not exist

Question 10 · from 1.10 · Calculate

No calculator. A function $g$ satisfies $4 x - 4 \leq g (x) \leq x^{2}$ for all real numbers $x$ . Find $\lim_{x \to 2} g (x)$ .

Your answer

Question 11 · from 1.11 · Calculate

No calculator. Find $\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$ . Give an exact decimal.

Your answer

Question 12 · from 1.12 · Sort it

No calculator. A function $f$ is defined for all real numbers and is continuous at every real number except $x = - 1$ and $x = 3$ . Tap an interval, then tap whether $f$ is continuous on it.

Continuous on this interval

Not continuous on this interval

Question 13 · from 1.13 · Sort it

No calculator. Each card describes a function $f$ that is discontinuous at $x = c$ . Tap a card, then tap whether the discontinuity can be removed by defining or redefining $f (c)$ .

Removable by choosing $f (c)$

Not removable by any $f (c)$

Question 14 · from 1.14 · Decide and justify

No calculator. Let $q (x) = \frac{x^{2} - 5 x + 6}{x - 2}$ . A classmate says the graph of $q$ has a vertical asymptote at $x = 2$ , because the denominator is 0 there. Is the classmate right?

Which reason supports your answer?

Question 15 · from 1.15 · Calculate

No calculator. Find $\lim_{x \to \infty} \frac{5 x^{3} - 2 x}{4 - 2 x^{3}}$ . Give your answer as an exact decimal.

Your answer

0 of 15 answered.