Limits & Continuity Flashcards

(87 cards)

1
Q
# _define_:
limit
A

The
value that a
function approaches as the
input approaches
some value.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

How would you

  • *read** this
  • *aloud**?
A

The

  • *limit** as
  • x* approaches c of
  • *f(x)**
  • *equals L.**
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

In plainspeak,
what do
limits do?

A

Describe how

  • *functions behave**
  • *near a point.**
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

What is a

  • *reasonable estimate** for the
  • *limit of g(x)** as:

x→3?

A

As x→3, the limit of g(x)
does not exist
(g(x) is defined; but there is
no limit because there is
no finite value that g(x) approaches)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

What is a

  • *reasonable estimate** for the
  • *limit of g(x)** as:

x→5?

A

As x→5, the limit of g(x)
approaches approximately 4.2
(g(x) is undefined; and there is a
limit because there is a
finite value that g(x) approaches)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

What is a

  • *reasonable estimate** for the
  • *limit of g(x)** as:

x→7?

A

As x→7, the limit of g(x)
approaches 4
(g(x) is defined; and there is a
limit because there is a
finite value that g(x) approaches)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Is this
possible?

  • *g(x) is defined** at (x, g(x)); the
  • *limit exists** at that point; and the
  • *limit equals g(x)** at that point?
A

Yes.

See the limit of g(x) below as x→8.

The function value and the limit can be the same, although the
function value is irrelevant to finding the limit.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

Is this
possible?

  • *g(x) is defined** at (x, g(x)); the
  • *limit exists** at that point; and the
  • *limit does not equal g(x)** at that point?
A

Yes.

See the limit of g(x) below as x→7.

The function value and the limit can be different because the
function value is irrelevant to finding the limit.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Is this
possible?

g(x) is defined at (x, g(x))
and the
limit does not exist at that point?

A

Yes.

  • See the limit of g(x) below as x→3.*
  • Just because the function is defined does not mean that the limit exists.*
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Is this
possible?

g(x) is undefined at an x value
and the
limit exists at that x value?

A

Yes.

  • See the limit of g(x) below as x→5.*
  • Just because the function is undefined does not mean that the limit does not exist.​*
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

What a

  • *reasonable estimation** of the
  • *limit** of the function below as
  • *x→2**?
A

The
limit does not exist.

We don’t say “unbounded” because the function is not approaching a finite value and it does not go in the same direction as x→2**.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

What a

  • *reasonable estimation** of the
  • *limit** of the function below as
  • *x→2**?
A

The
limit is unbounded.

The limit does not exist, but we say “unbounded” because the function is not approaching a finite value, although the two sides are going in the same direction.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

What
kind of limit
is this?

A

A
one-sided limit.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

How would you

  • *read** this
  • *aloud**?
A

The
limit of f(x) as x approaches 2
from the left.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

How would you

  • *read** this
  • *aloud**?
A

The
limit of f(x) as x approaches 2
from the right.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

limx→2+ f(x) = _____

A

0.5

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

limx→2 f(x) = _____

A

2

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

For

f(x) = 4x + 2,

where can you
evaluate the limit of
f(x)?

A

Anywhere f(x) is defined,
which is
anywhere.

For one given function, you can take the limit at an infinite number of points. Although we only usually care about limits near interesting points.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

Is this
possible?

  • *A sinusoid**,
  • *a line**, and
  • *a tangent graph** all have the
  • *same limit**?
A

Yes.

Functions that have the same limit at a point can look very different.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

Assuming this table is accurate,
is it
appropriate for
approximating
limx→2 f(x)
?

A

No, because the
increments are too large
approaching x = 2.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

Assuming this table is accurate,
is it
appropriate for
approximating
limx→2 f(x)
?

A
  • *No**, it approaches x = 2 from
  • *one side only**.
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

Assuming this table is accurate,
is it
appropriate for
approximating
limx→2 f(x)
?

A
  • *Yes**, it approaches x = 2 from
  • *both sides** at
  • *smaller and smaller increments**.
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

If you were
constructing a table to
approximate the limit below,
what would some
appropriate values be?

A

−6.9, −6.99, −6.999, −6.9999

  • Your values should approximate getting infinitely close to −7 from the right, which you do by approaching −7 from the right at smaller and smaller increments.*
  • k+ means “approaching k from the right,” whether those values are positive or negative.*
  • k means “approaching k from the left,” whether those values are positive or negative.*
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

What is a
reasonable estimate for
limx→5 g(x)?

A

3.68

Limit value is distinct from function value.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
*This makes sense because _limits are just discrete values_.* *Sometimes called the "_sum property of limits_."*
26
*This makes sense because _limits are just discrete values_.* *Sometimes called the "_difference property of limits_."*
27
*This makes sense because _limits are just discrete values_.* *Sometimes called the "_product property of limits_."*
28
*This makes sense because _limits are just discrete values_.* *Sometimes called the "_constant multiple property of limits_."*
29
*So long as M ≠ 0.* *This makes sense because _limits are just discrete values_.* *Sometimes called the "_quotient property of limits_."*
30
*This makes sense because _limits are just discrete values_.* *Sometimes called the "_exponent property of limits_."*
31
**limx→3 (g(x) + h(x)) =** \_\_\_\_\_?
**4**. ## Footnote lim (g(x) + h(x)) x→3 = lim g(x) + lim h(x) x→3 x→3 = (5) + (−1) = 4
32
* *limx→0 _h(x)_ =** \_\_\_\_\_? * *g(x)**
**Does not exist**. ## Footnote *Simplifies to 4/0, which is undefined.*
33
**limx→−2 (g(x) • h(x)) =** \_\_\_\_\_?
**0**. ## Footnote *Simplifies to 0/4, which is defined.*
34
**limx→−2 (f(x) + g(x)) =** \_\_\_\_\_?
**4**. * *lim f(x) does not exist* *x→−2* * *lim g(x) does not exist* *x→−2* * *BUT lim (f(x) + g(x)) x→−2 can exist* _*IFF* *the two one-sided limits approach the same number*_ * *And here . . .* * *​*lim (f(x) + g(x)) x→−2 = 1 + 3 = 4 * *​*lim (f(x) + g(x)) x→−2+ = 3 + 1 = 4 * *The one-sided limits approach the same number, so the limit exists*
35
**limx→1 (f(x) + g(x)) =** \_\_\_\_\_?
**Does not exist**. * *lim f(x) does not exist* *x→1* * *lim g(x) = 0* *x→1* * *lim (f(x) + g(x)) x→1 can exist* _*IFF* *the two one-sided limits approach the same number*_ * *And here . . .* * *​*lim (f(x) + g(x)) x→1 = 2 + 0 = 1 * *​*lim (f(x) + g(x)) x→1+ = −1 + 0 = −1 * *The one-sided limits approach different numbers, so the limit does not exist* *Different = does not exist*
36
**limx→1 (f(x) • g(x)) =** \_\_\_\_\_?
**0**. * *lim f(x) does not exist* *x→1* * *lim g(x) = 0* *x→1* * *BUT lim (f(x) • g(x)) x→1 can exist* _*IFF* *the two one-sided limits approach the same number*_ * *And here . . .* * *​*lim (f(x) • g(x)) x→1 = 1 • 0 = 0 * *​*lim (f(x) • g(x)) x→1+ = 3 • 0 = 0 * *The one-sided limits approach the same number, so the limit exists*
37
***u* and *v* are functions such that:** ## Footnote _FIRST_: **lim *v*(*x*) = *L* *x→c*** *(which means that the limit of the _internal function exists as x→c_)* _SECOND_: **lim *v*(*x*) = *u*(*L)* *x→L*** *(which means that the limit of the _external function is continuous at x = L_, and that the limit exists at x = L)*
38
**limx→2 f (g(x)) = \_\_\_\_\_**?
**2** * As x approaches the input value, the _internal function exists_ and equals 3 (*the discontinuity at this location for the internal function is irrelevant)* * At the location indicated by that value, the _external function is continuous_ and equals 2 * So the limit of this composite function equals 2
39
**limx→3 g (f(x)) = \_\_\_\_\_**?
**Does not exist** * As x approaches the input value, the _internal function exists_ and equals 2 * At the location indicated by that value, the _external function is not continuous_ * So the limit of this composite function does not exist
40
**limx→−1 f (h(x)) = \_\_\_\_\_**?
**Does not exist** * As x approaches the input value, the limit of the _internal function does not exist_ * So the limit of this composite function does not exist
41
**limx→−3 g (h(x)) = \_\_\_\_\_**?
**3** * As x approaches the input value, the _internal function exists_ and equals 1 (*the discontinuity at this location for the internal function is irrelevant)* * At the location indicated by that value, the _external function is continuous_ and equals 3 * So the limit of this composite function equals 3
42
**limx→−2 h (g(x)) = \_\_\_\_\_**?
**Does not exist** * As x approaches the input value, the limit of the _internal function equals 0_ *(the discontinuity at this location for the internal function is irrelevant)* * At the location indicated by that value, the _external function has a discontinuity_ *(the fact that the function is defined at this location is irrelevant)* * So the limit of this composite function does not exist
43
* *f(x)** is * *continuous** at * *x = a** * *iff** \_\_\_\_\_.
**limx→a f(x) = f(a)**
44
If **f(x)** is **continuous at x = a**, then **limx→a f(x) = \_\_\_\_\_**?
**f(a)** *ex: f(x) below is a quadratic, which are _continuous_ and (generally) _defined for all real numbers_.* *So the limit as x approaches a, the limit is f(a).* *Just plug it in.*
45
For _f(x) = cos (3 (x − 2)) + 3_, **limx→​a f(x) = \_\_\_\_\_**?
**f(a)** *Because sinusoids are _continuous_ and _defined for all real numbers_.* *Just plug it in.*
46
**limx→π tan(x) = \_\_\_\_\_**?
**0** *Generally, if the _six trig functions_ are _defined at a point_, the _limit equals the value_ at that point.*
47
**limx→π cot(x) = \_\_\_\_\_**?
**Does not exist** ``` lim cot(x) x→π ``` = lim cos(x) ÷ lim sin(x) x→π x→π = _1_ lim tan(x) x→π = _1_ 0 *So limit is undefined.* *Generally, if the _six trig functions_ are _defined at a point_, the _limit equals the value_ at that point.*
48
**limx→0 sec(x) = \_\_\_\_\_**?
**1** ``` lim sec(x) x→0 ``` = _1_ lim cos(x) x→0 = _1_ 1 = 1 *The secant function will be undefined anywhere the cosine function = 0.* *Generally, if the _six trig functions_ are _defined at a point_, the _limit equals the value_ at that point.*
49
**limx→0 csc(x) = \_\_\_\_\_**?
**Does not exist** ``` lim csc(x) x→0 ``` = _1_ lim sin(x) x→0 = _1_ 0 *So limit is undefined. The cosecant function will be undefined anywhere the sine function = 0.* *Generally, if the _six trig functions_ are _defined at a point_, the _limit equals the value_ at that point.*
50
f(x) = { 2x for 0 \< x _\<_ 1 { √x for x \> 1 **limx→0.5 f(x) = \_\_\_\_\_**?
**1**
51
f(x) = { 2x for 0 \< x _\<_ 1 { √x for x \> 1 **limx→1 f(x) = \_\_\_\_\_**?
**2**
52
f(x) = { 2x for 0 \< x _\<_ 1 { √x for x \> 1 **limx→1 f(x) = \_\_\_\_\_**?
**Does not exist** *The one-sided limits approach different values.* limx→1 f(x) ≠ limx→1+ f(x)
53
f(x) = { 2x for 0 \< x _\<_ 1 { √x for x \> 1 **limx→1+ f(x) = \_\_\_\_\_**?
**1**
54
In calculating *limx→a f(x)*, what is your **first step**?
**Direct substitution.** * Try to evaluate the function directly.* * Just plug a into the function.*
55
In calculating *limx→a f(x)*, what are some **possible outcomes after direct substitution**?
* **Limit found** (probably): * _f(a) = b_, where b is a real number * _must inspect graph or table_ to learn more about x = a * **Asymptote** (probably): * _f(a) = b/0_*,* where b ≠ 0 * **Indeterminate form**: * _f(a) = 0/0_
56
_After attempting direct substitution_ on this limit: **limx→8 x1/3**, **what have you learned**?
**Limit found.** * After attempting direct substitution,* * f(a) = b* *where b is a real number is probably the limit.*
57
_After attempting direct substitution_ on this limit: * *limx→−1 _2x + 2_** * *x2 − 1** **what have you learned**?
**Indeterminate form.** *After attempting direct substitution,* *f(a) = _0_ 0* *means that the function isn't defined at f(a), but even so, there may still be a limit.*
58
_After attempting direct substitution_ on this limit: * *limx→5 _2x_** * *x − 5** **what have you learned**?
**Does not exist**, but is an **asymptote** (probably) ## Footnote *After attempting direct substitution,* *f(a) = _b_ 0* *where b ≠ 0 is probably an asymptote.*
59
In calculating *limx→a f(x)*, if _direct substitution_ leads to an _indeterminate form_, **what techniques might help**?
**Try to rewrite in an equivalent form**. * **Factoring**: * limx→−1 _x2 − x − 2_ x2 − 2x − 3 can be reduced to * limx→−1 _x − 2_ x − 3 by factoring and cancelling. * **Conjugates**: * limx→4 _√(x) − 2_ x − 4 can be rewritten as * limx→4 _1_ √(x) + 2 using conjugates and cancelling. * **Trig identities**: * limx→0 _sin(x)_ sin(2x) can be rewritten as * limx→0 _1_ 2cos(x) using a trig identity
60
When using **direct substitution** to calculate a **limit** and encountering an **indeterminate form**, how can **rewriting the expression help**?
It might **eliminate** whatever is causing the **denominator to equal 0**, which will yield a function ( g(x) ) that is **defined where f(a) is undefined**, but is otherwise **equivalent**, meaning that **limx→a f(x) = limx→a g(x)**. ## Footnote *e.g.* limx→−1 _x2 − x − 2_ x2 − 2x − 3 f(x) = _x2 − x − 2_ x2 − 2x − 3 g(x) = _x2 − x − 2_ x2 − 2x − 3 = _(x − 2)(x + 1)_ (x − 3)(x + 1) = _x − 2_ , x ≠ −1 x − 3 f(x) = g(x), x ≠ −1
61
In calculating *limx→a f(x)*, if _direct substitution_ and _rewriting the function fail_, leads to an **what might you do**?
**Approximate** using graphs and/or tables.
62
_Mathematically_, what is the **squeeze theorem**?
If * *f(x) _\<_ g(x) _\<_ h(x)**, * *limx→c f(x) = L**, and * *limx→c h(x) = L**, then **limx→c g(x) = L**. *g(x) is squeezed (or sandwiched) between the other two functions.* *Can be useful for wacky functions.*
63
Does it seem that you can use the **squeeze theorem** here?
**Yes.**
64
Does it seem that you can use the **squeeze theorem** here?
**No**, because the _limits of the other two functions_ at the location in question _must be equal_ to apply the squeeze theorem.
65
Does it seem that you can use the **squeeze theorem** here?
**No**, because _one function must be always below *f*_ and _one function must be always above *f*_ for _*x*-values near the intersection_.
66
_Mathematically_, how do you know whether **f(x) is continuous at x = c**?
f(x) is continuous at x = c * *iff** * *limx→c f(x) = f(c)** ## Footnote *Practically speaking, this means that you can graph the function at that location without picking up your pencil.*
67
Is this graph **continuous**? If not, what **type of discontinuity** is this?
**Not continuous** due to **point discontinuities**.
68
Is this graph **continuous**? If not, what **type of discontinuity** is this?
**Not continuous** due to **jump discontinuities**.
69
Is this graph **continuous**? If not, what **type of discontinuity** is this?
**Not continuous** due to **asymptotic or infinite discontinuities**.
70
_Analytically_, given limx→c f(x), how can you * *recognize** that a * *point discontinuity exists**?
**The limit exists** at x = c, but it **is not f(c)**. limx→c f(x) = L ≠ f(c)
71
_Analytically_, given limx→c f(x), how can you * *recognize** that a * *jump discontinuity exists**?
**The limit does not exist,** but the **one-sided limits do exist**. limx→c f(x) = L limx→c+ f(x) = M L ≠ M
72
_Analytically_, given limx→c f(x), how can you * *recognize** that an * *asymptotic discontinuity exists**?
**The limit does not exist,** and the **one-sided limits are unbounded**. limx→c f(x) = L limx→c+ f(x) = M L ≠ M
73
g(x) = { x3 for x \< 1 { 1 for x _\>_ 1 Is g(x) * *continuous** at * *x = 1**? _Analytically_, how do you **know**?
**It's continuous**. Both **one-sided limits** **exist** and are **equivalent**. limx→1 = g(1) = 1 = limx→1+
74
g(x) = { ln(x) for 0 \< x _\<_ 2 { x•ln(x) for x _\>_ 2 Is g(x) * *continuous** at * *x = 2**? _Analytically_, how do you **know**?
**It's not continuous**. both **one-sided limits** **exist**, but they **aren't equivalent**. limx→2 ≠ limx→2+
75
Which of * *functions *f* and *g*** (if either), is * *continuous over (−2, 3)**?
* **g* is continuous**; * **f* is not**. h(x) is continuous over (a, b) iff h(x) is continuous over every point in the interval.
76
Which of * *functions *f* and *g*** (if either), is * *continuous over [−2, 1]**?
* **f* is continuous**; * **g* is not**. h(x) is continuous over [a, b] iff h(x) is continuous over every point in the interval and limx→a+ h(x) = h(a) = limx→a h(x)
77
How do you * *remove** a * *point discontinuity**?
Plug it with **whatever the limit is** at that point.
78
Which limit **expressions agree** with the **graph**? ## Footnote A) limx→4 h(x) = −∞ B) limx→4+ h(x) = −∞ C) limx→−2 h(x) = −∞
**B** & **C** are correct: ## Footnote A) limx→4 h(x) = −∞ **B) limx→4+ h(x) = −∞** **C) limx→−2 h(x) = −∞**
79
**h(x) = _−5_ x** How would you * *describe** the * *one-sided limits** of h at * *x = 0**?
* *limx→0 = ∞** * (signs will be +/+ = +)* and * *limx→0+ = −∞** * (signs will be +/− = −)* *The easiest way to do this is with a weird function is to use a calculator to approach 0 incrementally.*
80
What are the **limits** of this function as **x approaches infinity** and **x approaches negative infinity?**
**limx→−∞ g(x) = 4** and **limx→∞ g(x) = 1**
81
Which graph _agrees_ with this statement? ## Footnote **limx→∞ = −2**
**B**
82
f(x) = _3x5 − 17x2 + 3_ 6x5 − 100x2 − 10 **limx→∞** **f(x) = \_\_\_\_\_**?
**_1_ 2** *At _increasingly extreme values of x_, the _highest-degree terms_ will _increasingly dominate_ the value.* f(x) = _3x5 − 17x2 + 3_ 6x5 − 100x2 − 10 ≈ _3x5​_ *(as x →∞)* 6x5 = 1/2 limx→∞ f(x) = 1/2
83
f(x) = _3x3 − 2x2 + 7_ 6x4 − x3 + 2x − 100 **limx→−∞** **f(x) = \_\_\_\_\_**?
**0** *At _increasingly extreme values of x_, the _highest-degree terms_ will _increasingly dominate_ the value.* f(x) = _3x3 − 2x2 + 7_ 6x4 − x3 + 2x − 100 ≈ _3x3​_ *(as x →−∞)* 6x4 = _1_ 2x *This will be 1 divided by an increasingly extreme, negative number, which will make it approach 0.* limx→−∞ f(x) = 0
84
f(x) = _√(9x2 + 2)_ 2x − 2 **limx→∞** **f(x) = \_\_\_\_\_**?
**3/2** *At _increasingly extreme values of x_, the _highest-degree terms_ will _increasingly dominate_ the value.* f(x) = _√(9x2)_ 2x ≈ _√(9x2​)_ *(as x →∞)* 2x = _3x_ *(positive value here)* 2x *(positive value here)* = _3_ 2 limx→∞ f(x) = 3/2
85
f(x) = _√(9x2 + 2)_ 2x − 2 **limx→−∞** **f(x) = \_\_\_\_\_**?
**−3/2** *At _increasingly extreme values of x_, the _highest-degree terms_ will _increasingly dominate_ the value.* f(x) = _√(9x2)_ 2x ≈ _√(9x2​)_ *(as x →−∞)* 2x = _3x_ *(positive value here)* 2x *(negative value here)* = _−3_ 2 limx→−∞ f(x) = −3/2
86
In _plainspeak_, describe the **intermediate value theorem**.
If **f(x) is continuous over [a, b]**, then FIRST: * *every value [a, b]** will have * *exactly one dance partner** and SECOND: * *every value [f(a), f(b)]** will have * *at least one dance partner**
87
_Mathematically_, describe the **intermediate value theorem**.
If **f(x) is continuous over [a, b]**, then FIRST: f(x) will **take every value between f(a) and f(b)** over the interval and SECOND: **for any L between f(a) and f(b)**, there is at least **one number c in [a, b]** for which **f(c) = L**