Study

Question 1

Why is lim sin3x / x = 0?

x—> ∞

Answer 1

Because lim sinx / x = 0,
x—> ∞
So multiply top and bottom by 3 and cancel.

Question 2

How do you find the limit at infinity for lim 4x^2 + x + 1 / 2x^3 - 5?
x—> ∞

Answer 2

Divide everything by the highest power of x, simplify, then replace x with ∞, which is same as saying 4/0 so they all become zero.

Question 3

Do this example:

lim 6-√x / 36x-x^2
x—>36

Answer 3

1/432

Question 4

Do this example:

lim t^2-4 / 2t^2 + 5t+2
x—>-2

Answer 4

4/3

Question 5

How is the the graph of a limit if lim f(x)=-1, lim f(x)=2, f(0)=1
x—>0- x—>0+

Answer 5

Open circle at y=2 and line to the right of it, closed circle at y=1, and open circle at y=-1 with line to the left of it

Question 6

If you are to find the limit coming from the left or right side on a graph, what is done?

Answer 6

Use the closet line to the point, and the first y-value is the limit.

Question 7

Use the closet line to the point, and the first y-value is the limit.

Answer 7

If you are to find the limit coming from the left or right side on a graph

Question 8

What does this mean?:

lim f(x) = -∞
x—>3+

Answer 8

The values of f(x) can be made negative with arbitrarily large absolute values by taking x sufficiently close to 3.

Question 9

The values of f(x) can be made negative with arbitrarily large absolute values by taking x sufficiently close to 3.

Answer 9

lim f(x) = -∞
x—>3+

Question 10

What does this mean?:

lim f(x) = ∞
x—>-4

Answer 10

The values of f(x) can be made arbitrarily large by taking x sufficiently close to -4.

Question 11

The values of f(x) can be made arbitrarily large by taking x sufficiently close to -4.

Answer 11

lim f(x) = ∞
x—>-4

Question 12

Do this example:

Find continuity and domain for:

f(x)=ln(x)+tan^-1(x) / x^2 -1

Answer 12

ln(x) continuous on (0, ∞)
tan^-1(x) continuous for (-∞, ∞)
x^2 -1 continuous for (-∞, ∞)

f(x) continuous on {A∩B∩C | x^2 -1 ≠ 0}

Domain: (0,1)U(1, ∞)

Question 13

List all continuous types of functions.

Answer 13

Polynomials, trigonometric, exponential, rational, inverse trigonometric, logarithmic, root.

Question 14

Polynomials, trigonometric, exponential, rational, inverse trigonometric, logarithmic, root.

Answer 14

All continuous types of functions.

Question 15

If f and g are continuous at a, and c is a constant, then which functions are also continuous?

Answer 15

f+g, f-g, cf, fg, f/g for g≠0. Can be proved with limit laws and hold on intervals.

Question 16

f+g, f-g, cf, fg, f/g for g≠0. Can be proved with limit laws and hold on intervals.

Answer 16

If f and g are continuous at a, and c is a constant, then these functions are also continuous.

Question 17

Do this example:

lim x^3 • cos(1/x)
x—>0

Answer 17

lim x^3 • cos(1/x)= 0

x—>0

Question 18

Do this example:

lim xsin(x)
x—>0

Answer 18

lim xsin(x)=0
x—>0

Question 19

Do this example:

5x+2<=f(x)<=x^2 + 8 as x—>2

Answer 19

lim f(x) = 12
x—>2

Question 20

Do this example:

Where is this function continuous?:

f(x)=x^2 -3x-10 / x^2 -8x+15

Answer 20

f(x) = {x+2 / x-3 if x≠5,3 DNE if x=5,3}

Question 21

How do you check if
f(x)=1-√1-x^2 is continuous on
x—>a
the interval [-1,1]?

Answer 21

Check if check if f(x) = f(a)

Question 22

Check if check if f(x) = f(a)

Answer 22

When checking if continuous on an interval or a point.

Question 23

lim f(x) and lim f(x) = what?
x—>a-         x—>a+

Answer 23

f(a)

Question 24

Draw the graph of [|x|], then determine when it is discontinuous.

Answer 24

Discontinuous when approaching from left, as it is a different y-value.

Question 25

When is a function discontinuous?

Answer 25

When you get zero in the denominator, or it is equal to infinity.

Question 26

Discontinuous when approaching from left, as it is a different y-value.

Answer 26

[|x|]

Question 27

Equal to f(a)

Answer 27

lim f(x) and lim f(x)
x—>a-         x—>a+

Question 28

When checking if continuous on an interval or a point.

Answer 28

Check if check if f(x) = f(a)

Question 29

When you get zero in the denominator, or it is equal to infinity.

Answer 29

When a function is discontinuous

Question 30

Do this example:

lim |x-2| / x-2

Answer 30

|x-2| = { x-2, x >=2 -(x-2), x< 2}

Find limit from both of these x-values and compare. If not same, limit DNE.

Question 31

Do this example:

lim tanx / x
x—>0

Answer 31

sinx/cosx / x

Get sinx/x by itself because it equals 1.

Replace x with limit point and solve.

Question 32

Do this example:

lim 4sin5x / sin4x

Answer 32

Multiply top and bottom by (5x) and (x) at the same time. Cancel 4x/sin4x and sin5x/5x.

Question 33

Do this example:

lim x+sinx / x

Answer 33

= 1 + sinx/x = 1+1 = 2

Question 34

If a limit equation is factored and still gives 0 in denominator.

Answer 34

Infinite limit.

Question 35

When asked to find the limit on the graph, with no direction, and there is multiple x-values there, what does this mean?

Answer 35

Discontinuous

Question 36

The graph of a limit is discontinuous when.

Answer 36

There is no direction given, and there is multiple of the same x-values.

Question 37

A continuous function has no

Answer 37

Breaks

Question 38

Has no breaks

Answer 38

Continuous function

Question 39

Why can’t the product rule be used on lim x^2 sin(1/x)?

x—>0

Answer 39

Because 0 in the denominator DNE.

Question 40

If f(x) <= g(x) when x is near a, and the limits of f and g both exist as x approaches a, then

Answer 40

lim f(x) <= lim g(x)
x—>a         x—>a

Question 41

lim f(x) <= lim g(x) when
x—>a         x—>a

Answer 41

f(x) <= g(x) when x is near a and the limits of both functions exist as x approaches a.

Question 42

Do this example:

lim √t^2 +9 -3 / t^2
x—> 0

Answer 42

Quotient rule does not work (taking limits of top and bottom), because denominator = 0, so:

Multiply by reciprocal. Simplify, and place 0 into x.

Question 43

Do this example:

lim x^3 +2x^2 -1 / 5-3x
x—> -2

Answer 43

Find limit of all x’s and solve.

Question 44

lim[f(x)+g(x)] equals

x—>a

Answer 44

lim f(x) + lim g(x)
x—>a      x—>a

Question 45

lim f(x) + lim g(x) equals
x—>a      x—>a

Answer 45

lim[f(x)+g(x)]

x—>a

Question 46

lim[c f(x)] equals

x—>a

Answer 46

c lim f(x)

x—>a

Question 47

c lim f(x) equals

x—>a

Answer 47

lim[c f(x)]

x—>a

Question 48

lim(f(x))^n equals

x—>a

Answer 48

lim(f(x))^n-1 • lim f(x)
x—>a x—>a
which equals:

[lim f(x)]^n
x—>a