Limits & Continuity

Question 1

# _define_:
limit

Answer 1

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

Question 2

How would you

• *read** this
• *aloud**?

Answer 2

The

• *limit** as
• x* approaches c of
• *f(x)**
• *equals L.**

Question 3

In plainspeak,
what do
limits do?

Answer 3

Describe how

• *functions behave**
• *near a point.**

Question 4

What is a

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

x→3?

Answer 4

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)

Question 5

What is a

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

x→5?

Answer 5

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)

Question 6

What is a

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

x→7?

Answer 6

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)

Question 7

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?

Answer 7

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.

Question 8

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?

Answer 8

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.

Question 9

Is this
possible?

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

Answer 9

Yes.

• See the limit of g(x) below as x→3.*
• Just because the function is defined does not mean that the limit exists.*

Question 10

Is this
possible?

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

Answer 10

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.​*

Question 11

What a

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

Answer 11

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**.

Question 12

What a

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

Answer 12

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.

Question 13

What
kind of limit
is this?

Answer 13

A
one-sided limit.

Question 14

How would you

• *read** this
• *aloud**?

Answer 14

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

Question 15

How would you

• *read** this
• *aloud**?

Answer 15

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

Question 16

lim_x→2⁺ f(x) = _____

Answer 16

0.5

Question 17

lim_x→2⁻ f(x) = _____

Answer 17

2

Question 18

For

f(x) = 4x + 2,

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

Answer 18

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.

Question 19

Is this
possible?

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

Answer 19

Yes.

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

Question 20

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

Answer 20

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

Question 21

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

Answer 21

• *No**, it approaches x = 2 from
• *one side only**.

Question 22

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

Answer 22

• *Yes**, it approaches x = 2 from
• *both sides** at
• *smaller and smaller increments**.

Question 23

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

Answer 23

−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.*

Question 24

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

Answer 24

3.68

Limit value is distinct from function value.

Question 25

Answer 25

*This makes sense because _limits are just discrete values_.* *Sometimes called the "_sum property of limits_."*

Question 26

Answer 26

*This makes sense because _limits are just discrete values_.* *Sometimes called the "_difference property of limits_."*

Question 27

Answer 27

*This makes sense because _limits are just discrete values_.* *Sometimes called the "_product property of limits_."*

Question 28

Answer 28

*This makes sense because _limits are just discrete values_.* *Sometimes called the "_constant multiple property of limits_."*

Question 29

Answer 29

*So long as M ≠ 0.* *This makes sense because _limits are just discrete values_.* *Sometimes called the "_quotient property of limits_."*

Question 30

Answer 30

*This makes sense because _limits are just discrete values_.* *Sometimes called the "_exponent property of limits_."*

Question 31

**lim_x→3 (g(x) + h(x)) =** \_\_\_\_\_?

Answer 31

**4**. ## Footnote lim (g(x) + h(x)) ^x→3 = lim g(x) + lim h(x) ^{x→3 x→3} = (5) + (−1) = 4

Question 32

* *lim_x→0 _h(x)_ =** \_\_\_\_\_? * *g(x)**

Answer 32

**Does not exist**. ## Footnote *Simplifies to 4/0, which is undefined.*

Question 33

**lim_x→−2 (g(x) • h(x)) =** \_\_\_\_\_?

Answer 33

**0**. ## Footnote *Simplifies to 0/4, which is defined.*

Question 34

**lim_x→−2 (f(x) + g(x)) =** \_\_\_\_\_?

Answer 34

**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*

Question 35

**lim_x→1 (f(x) + g(x)) =** \_\_\_\_\_?

Answer 35

**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*

Question 36

**lim_x→1 (f(x) • g(x)) =** \_\_\_\_\_?

Answer 36

**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*

Question 37

Answer 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)*

Question 38

**lim_x→2 f (g(x)) = \_\_\_\_\_**?

Answer 38

**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

Question 39

**lim_x→3 g (f(x)) = \_\_\_\_\_**?

Answer 39

**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

Question 40

**lim_x→−1 f (h(x)) = \_\_\_\_\_**?

Answer 40

**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

Question 41

**lim_x→−3 g (h(x)) = \_\_\_\_\_**?

Answer 41

**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

Question 42

**lim_x→−2 h (g(x)) = \_\_\_\_\_**?

Answer 42

**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

Question 43

* *f(x)** is * *continuous** at * *x = a** * *iff** \_\_\_\_\_.

Answer 43

**lim_x→a f(x) = f(a)**

Question 44

If **f(x)** is **continuous at x = a**, then **lim_x→a f(x) = \_\_\_\_\_**?

Answer 44

**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.*

Question 45

For _f(x) = cos (3 (x − 2)) + 3_, **lim_x→​a f(x) = \_\_\_\_\_**?

Answer 45

**f(a)** *Because sinusoids are _continuous_ and _defined for all real numbers_.* *Just plug it in.*

Question 46

**lim_x→π tan(x) = \_\_\_\_\_**?

Answer 46

**0** *Generally, if the _six trig functions_ are _defined at a point_, the _limit equals the value_ at that point.*

Question 47

**lim_x→π cot(x) = \_\_\_\_\_**?

Answer 47

**Does not exist** ``` lim cot(x) <sup>x→π</sup> ``` = 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.*

Question 48

**lim_x→0 sec(x) = \_\_\_\_\_**?

Answer 48

**1** ``` lim sec(x) <sup>x→0</sup> ``` = _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.*

Question 49

**lim_x→0 csc(x) = \_\_\_\_\_**?

Answer 49

**Does not exist** ``` lim csc(x) <sup>x→0</sup> ``` = _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.*

Question 50

f(x) = { 2x for 0 \< x _\<_ 1 { √x for x \> 1 **lim_x→0.5 f(x) = \_\_\_\_\_**?

Answer 50

**1**

Question 51

f(x) = { 2x for 0 \< x _\<_ 1 { √x for x \> 1 **lim_x→1⁻ f(x) = \_\_\_\_\_**?

Answer 51

**2**

Question 52

f(x) = { 2x for 0 \< x _\<_ 1 { √x for x \> 1 **lim_x→1 f(x) = \_\_\_\_\_**?

Answer 52

**Does not exist** *The one-sided limits approach different values.* lim_x→1⁻ f(x) ≠ lim_x→1⁺ f(x)

Question 53

f(x) = { 2x for 0 \< x _\<_ 1 { √x for x \> 1 **lim_x→1⁺ f(x) = \_\_\_\_\_**?

Answer 53

**1**

Question 54

In calculating *lim_x→a f(x)*, what is your **first step**?

Answer 54

**Direct substitution.** * Try to evaluate the function directly.* * Just plug a into the function.*

Question 55

In calculating *lim_x→a f(x)*, what are some **possible outcomes after direct substitution**?

Answer 55

* **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_

Question 56

_After attempting direct substitution_ on this limit: **lim_x→8 x^1/3**, **what have you learned**?

Answer 56

**Limit found.** * After attempting direct substitution,* * f(a) = b* *where b is a real number is probably the limit.*

Question 57

_After attempting direct substitution_ on this limit: * *lim_x→−1 _2x + 2_** * *x² − 1** **what have you learned**?

Answer 57

**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.*

Question 58

_After attempting direct substitution_ on this limit: * *lim_x→5 _2x_** * *x − 5** **what have you learned**?

Answer 58

**Does not exist**, but is an **asymptote** (probably) ## Footnote *After attempting direct substitution,* *f(a) = _b_ 0* *where b ≠ 0 is probably an asymptote.*

Question 59

In calculating *lim_x→a f(x)*, if _direct substitution_ leads to an _indeterminate form_, **what techniques might help**?

Answer 59

**Try to rewrite in an equivalent form**. * **Factoring**: * lim_x→−1 _x² − x − 2_ x² − 2x − 3 can be reduced to * lim_x→−1 _x − 2_ x − 3 by factoring and cancelling. * **Conjugates**: * lim_x→4 _√(x) − 2_ x − 4 can be rewritten as * lim_x→4 _1_ √(x) + 2 using conjugates and cancelling. * **Trig identities**: * lim_x→0 _sin(x)_ sin(2x) can be rewritten as * lim_x→0 _1_ 2cos(x) using a trig identity

Question 60

When using **direct substitution** to calculate a **limit** and encountering an **indeterminate form**, how can **rewriting the expression help**?

Answer 60

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 **lim_x→a f(x) = lim_x→a g(x)**. ## Footnote *e.g.* lim_x→−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

Question 61

In calculating *lim_x→a f(x)*, if _direct substitution_ and _rewriting the function fail_, leads to an **what might you do**?

Answer 61

**Approximate** using graphs and/or tables.

Question 62

_Mathematically_, what is the **squeeze theorem**?

Answer 62

If * *f(x) _\<_ g(x) _\<_ h(x)**, * *lim_x→c f(x) = L**, and * *lim_x→c h(x) = L**, then **lim_x→c g(x) = L**. *g(x) is squeezed (or sandwiched) between the other two functions.* *Can be useful for wacky functions.*

Question 63

Does it seem that you can use the **squeeze theorem** here?

Answer 63

**Yes.**

Question 64

Does it seem that you can use the **squeeze theorem** here?

Answer 64

**No**, because the _limits of the other two functions_ at the location in question _must be equal_ to apply the squeeze theorem.

Question 65

Does it seem that you can use the **squeeze theorem** here?

Answer 65

**No**, because _one function must be always below *f*_ and _one function must be always above *f*_ for _*x*-values near the intersection_.

Question 66

_Mathematically_, how do you know whether **f(x) is continuous at x = c**?

Answer 66

f(x) is continuous at x = c * *iff** * *lim_x→c f(x) = f(c)** ## Footnote *Practically speaking, this means that you can graph the function at that location without picking up your pencil.*

Question 67

Is this graph **continuous**? If not, what **type of discontinuity** is this?

Answer 67

**Not continuous** due to **point discontinuities**.

Question 68

Is this graph **continuous**? If not, what **type of discontinuity** is this?

Answer 68

**Not continuous** due to **jump discontinuities**.

Question 69

Is this graph **continuous**? If not, what **type of discontinuity** is this?

Answer 69

**Not continuous** due to **asymptotic or infinite discontinuities**.

Question 70

_Analytically_, given lim_x→c f(x), how can you * *recognize** that a * *point discontinuity exists**?

Answer 70

**The limit exists** at x = c, but it **is not f(c)**. lim_x→c f(x) = L ≠ f(c)

Question 71

_Analytically_, given lim_x→c f(x), how can you * *recognize** that a * *jump discontinuity exists**?

Answer 71

**The limit does not exist,** but the **one-sided limits do exist**. lim_x→c⁻ f(x) = L lim_x→c⁺ f(x) = M L ≠ M

Question 72

_Analytically_, given lim_x→c f(x), how can you * *recognize** that an * *asymptotic discontinuity exists**?

Answer 72

**The limit does not exist,** and the **one-sided limits are unbounded**. lim_x→c⁻ f(x) = L lim_x→c⁺ f(x) = M L ≠ M

Question 73

g(x) = { x³ for x \< 1 { 1 for x _\>_ 1 Is g(x) * *continuous** at * *x = 1**? _Analytically_, how do you **know**?

Answer 73

**It's continuous**. Both **one-sided limits** **exist** and are **equivalent**. lim_x→1⁻ = g(1) = 1 = lim_x→1⁺

Question 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**?

Answer 74

**It's not continuous**. both **one-sided limits** **exist**, but they **aren't equivalent**. lim_x→2⁻ ≠ lim_x→2⁺

Question 75

Which of * *functions *f* and *g*** (if either), is * *continuous over (−2, 3)**?

Answer 75

* **g* is continuous**; * **f* is not**. h(x) is continuous over (a, b) iff h(x) is continuous over every point in the interval.

Question 76

Which of * *functions *f* and *g*** (if either), is * *continuous over [−2, 1]**?

Answer 76

* **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 lim_x→a⁺ h(x) = h(a) = lim_x→a⁻ h(x)

Question 77

How do you * *remove** a * *point discontinuity**?

Answer 77

Plug it with **whatever the limit is** at that point.

Question 78

Which limit **expressions agree** with the **graph**? ## Footnote A) lim_x→4 h(x) = −∞ B) lim_x→4⁺ h(x) = −∞ C) lim_x→−2 h(x) = −∞

Answer 78

**B** & **C** are correct: ## Footnote A) lim_x→4 h(x) = −∞ **B) lim_x→4⁺ h(x) = −∞** **C) lim_x→−2 h(x) = −∞**

Question 79

**h(x) = _−5_ x** How would you * *describe** the * *one-sided limits** of h at * *x = 0**?

Answer 79

* *lim_x→0⁻ = ∞** * (signs will be +/+ = +)* and * *lim_x→0⁺ = −∞** * (signs will be +/− = −)* *The easiest way to do this is with a weird function is to use a calculator to approach 0 incrementally.*

Question 80

What are the **limits** of this function as **x approaches infinity** and **x approaches negative infinity?**

Answer 80

**lim_x→−∞ g(x) = 4** and **lim_x→∞ g(x) = 1**

Question 81

Which graph _agrees_ with this statement? ## Footnote **lim_x→∞ = −2**

Answer 81

**B**

Question 82

f(x) = _3x⁵ − 17x² + 3_ 6x⁵ − 100x² − 10 **lim_x→∞** **f(x) = \_\_\_\_\_**?

Answer 82

**_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 ≈ _3x^5​_ *(as x →∞)* 6x⁵ = 1/2 lim_x→∞ f(x) = 1/2

Question 83

f(x) = _3x³ − 2x² + 7_ 6x⁴ − x³ + 2x − 100 **lim_x→−∞** **f(x) = \_\_\_\_\_**?

Answer 83

**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 ≈ _3x^3​_ *(as x →−∞)* 6x⁴ = _1_ 2x *This will be 1 divided by an increasingly extreme, negative number, which will make it approach 0.* lim_x→−∞ f(x) = 0

Question 84

f(x) = _√(9x² + 2)_ 2x − 2 **lim_x→∞** **f(x) = \_\_\_\_\_**?

Answer 84

**3/2** *At _increasingly extreme values of x_, the _highest-degree terms_ will _increasingly dominate_ the value.* f(x) = _√(9x²)_ 2x ≈ _√(9x^2​)_ *(as x →∞)* 2x = _3x_ *(positive value here)* 2x *(positive value here)* = _3_ 2 lim_x→∞ f(x) = 3/2

Question 85

f(x) = _√(9x² + 2)_ 2x − 2 **lim_x→−∞** **f(x) = \_\_\_\_\_**?

Answer 85

**−3/2** *At _increasingly extreme values of x_, the _highest-degree terms_ will _increasingly dominate_ the value.* f(x) = _√(9x²)_ 2x ≈ _√(9x^2​)_ *(as x →−∞)* 2x = _3x_ *(positive value here)* 2x *(negative value here)* = _−3_ 2 lim_x→−∞ f(x) = −3/2

Question 86

In _plainspeak_, describe the **intermediate value theorem**.

Answer 86

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**

Question 87

_Mathematically_, describe the **intermediate value theorem**.

Answer 87

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**