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.

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.

Simplifies to 4/0, which is undefined.

Question 33

lim_x→−2 (g(x) • h(x)) = _____?

Answer 33

0.

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^{<span>+</span>}
    = 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^{<span>+</span>}
    = −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^{<span>+</span>}
    = 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:

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)

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).

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)**

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^{<span>−</span>}} f(x) = L

lim_{x→c^{<span>+</span>}} 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^{<span>−</span>}} f(x) = L

lim_{x→c^{<span>+</span>}} 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?

A) lim_x→4 h(x) = −∞

B) lim_x→4⁺ h(x) = −∞

C) lim_x→−2 h(x) = −∞

Answer 78

B & C
are correct:

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?

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