Lesson test: Limits

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Limits

1.[3p]
In the definition of $lim_{x \to a} f (x) = L$ , which order do the two tolerances come in?

The output tolerance is named first, and the input tolerance may depend on it

Correct

The input tolerance is named first, and the output tolerance may depend on it

The answer is: The output tolerance is named first, and the input tolerance may depend on it

Both are named at once and must be equal

The answer is: The output tolerance is named first, and the input tolerance may depend on it

Neither depends on the other

The answer is: The output tolerance is named first, and the input tolerance may depend on it

2.[2p]
The value of $lim_{x \to a} f (x)$ can be found without knowing anything about $f (a)$ .

True

Correct

False

The answer is: True

3.[2p]
Evaluate $lim_{x \to 2} \frac{x^{3} - 8}{x - 2}$ .

CorrectNot quite: 12

4.[3p]
Evaluate $lim_{h \to 0} \frac{\sqrt{4 + h} - 2}{h}$ .

CorrectNot quite: 0.25

5.[2p]
Evaluate $lim_{x \to \infty} \frac{3 x^{2} + 2 x - 1}{5 x^{2} - x + 4}$ .

CorrectNot quite: 0.6

6.[2p]
Why is $0 / 0$ called an indeterminate form?

Because the form alone determines nothing: limits of that shape can come out to any value or to none

Correct

Because the answer is always zero

The answer is: Because the form alone determines nothing: limits of that shape can come out to any value or to none

Because the answer can never be found

The answer is: Because the form alone determines nothing: limits of that shape can come out to any value or to none

Because the limit always fails to exist

The answer is: Because the form alone determines nothing: limits of that shape can come out to any value or to none

7.[3p]
Match each limit as $x$ approaches $0$ to how it behaves.

$| x | / x$
$1 / x^{2}$
$\sin (1 / x)$
$x^{2} \sin (1 / x)$

the limit is zero, by squeezing
the values oscillate forever
the values exceed every bound
the two sides disagree

Show the answer

$| x | / x$ : the two sides disagree $1 / x^{2}$ : the values exceed every bound $\sin (1 / x)$ : the values oscillate forever $x^{2} \sin (1 / x)$ : the limit is zero, by squeezing

8.[3p]
Which of these are correct?

Select all that apply

$lim_{x \to a} P (x) = P (a)$ for every polynomial $P$

Correct

Writing $lim_{x \to 0} 1 / x^{2} = \infty$ is shorthand for a way of not existing

Correct

A two-sided limit exists exactly when both one-sided limits exist and agree

Correct

If a limit is an indeterminate form, the limit does not exist

The answer is: $lim_{x \to a} P (x) = P (a)$ for every polynomial $P$ , Writing $lim_{x \to 0} 1 / x^{2} = \infty$ is shorthand for a way of not existing, A two-sided limit exists exactly when both one-sided limits exist and agree

9.[2p]
The squeeze theorem concludes that $f$ has limit $L$ at $a$ when

$f$ is trapped between two functions that both have limit $L$ there

Correct

$f$ is bounded near $a$

The answer is: $f$ is trapped between two functions that both have limit $L$ there

$f$ is continuous at $a$

The answer is: $f$ is trapped between two functions that both have limit $L$ there

$f$ has equal one-sided limits at $a$

The answer is: $f$ is trapped between two functions that both have limit $L$ there

1.[3p] In the definition of limx→af(x)=L, which order do the two tolerances come in?

2.[2p] The value of limx→af(x) can be found without knowing anything about f(a).

3.[2p] Evaluate limx→2x3−8x−2.

4.[3p] Evaluate limh→04+h−2h.

5.[2p] Evaluate limx→∞3x2+2x−15x2−x+4.

6.[2p] Why is 0/0 called an indeterminate form?

7.[3p] Match each limit as x approaches 0 to how it behaves.

8.[3p] Which of these are correct?

9.[2p] The squeeze theorem concludes that f has limit L at a when

1.[3p]
In the definition of $lim_{x \to a} f (x) = L$ , which order do the two tolerances come in?

2.[2p]
The value of $lim_{x \to a} f (x)$ can be found without knowing anything about $f (a)$ .

3.[2p]
Evaluate $lim_{x \to 2} \frac{x^{3} - 8}{x - 2}$ .

4.[3p]
Evaluate $lim_{h \to 0} \frac{\sqrt{4 + h} - 2}{h}$ .

5.[2p]
Evaluate $lim_{x \to \infty} \frac{3 x^{2} + 2 x - 1}{5 x^{2} - x + 4}$ .

6.[2p]
Why is $0 / 0$ called an indeterminate form?

7.[3p]
Match each limit as $x$ approaches $0$ to how it behaves.

8.[3p]
Which of these are correct?

9.[2p]
The squeeze theorem concludes that $f$ has limit $L$ at $a$ when