Unit 1 - Limits & Continuity

Question 1

What is the secant line?

Answer 1

Straight line that connects two points on a function; equivalent to the AVERAGE ROC

Question 2

What is the tangent line?

Answer 2

Straight line that connects two very close points or one point on a function; equivalent to the INSTANTANEOUS ROC

Question 3

How do you read a limit?

Answer 3

“The limit x approaches (x-value) of f(x) is …”

Question 4

T or F: Limit and y-value could be the same value but does not have to be the same value.

Answer 4

True

Question 5

What information does a limit give you?

Answer 5

A limit tells you what point a function is APPROACHING , but it doesn’t tell you what the VALUE of that particular point is.

Question 6

What is a one-sided limit?

Answer 6

A limit in which you define the y-value of a function as it approaches a given x-value from either the left or right side

Question 7

How do you notate a limit from the left side?

Answer 7

Add a NEGATIVE sign after the x-value

Question 8

How do you notate a limit from the right side?

Answer 8

Add a POSITIVE sign after the x-value

Question 9

How can you determine what value a limit is approaching from the table?

Answer 9

As the x-values get increasingly closer to the point of interest, the y-value will also increasingly approach a value.

Question 10

How do you evaluate a limit at a point?

Answer 10

direct substitution or factor and cancel

Question 11

If two sides of a limit does not meet, then ….

Answer 11

The limit DOES NOT exist

Question 12

lim   sin(x)/x = 
x->0

Answer 12

1

Question 13

lim (1 - cos(x))/x =

x->0

Answer 13

0

Question 14

If you completely factor the numerator and denominator of an expression and no factors cancel, then…

Answer 14

the limit DOES NOT exist

Question 15

When an indeterminate form appears, what strategies can you use to determine the limit

Answer 15

Complex fractions

Rationalizing with Radicals

Question 16

Squeeze Theorem

Answer 16

1- If g(x) <= f(x) <= h(x)
2- If lim g(x) = L and lim h(x) = L
       x->a                 x->a
Then lim f(x) = L
         x->a

Question 17

How do you solve absolute value limits?

Answer 17

Find numbers that are close to x to substitute as x

Question 18

Three conditions of continuity

Answer 18

1- f(c) is defined (with c=domain)
2- lim f(x) exists
    x->c
3- lim f(x) = f(c)
    x->c

Question 19

Three types of discontinuities

Answer 19

Hole/ Removable
Vertical Asymptote/ Non-removable
Jump/ Non-removable

Question 20

In a function, discontinuities are found when …

Answer 20

the denominator is equivalent to 0

Question 21

If the factors in an expression cancels, then the discontinuity can be classified as…

Answer 21

hole discontinuity

Question 22

If the factors in an expression does not cancels, then the discontinuity can be classified as…

Answer 22

an vertical asymptote

Question 23

If the left side of a limit is different from the right side of a limit, then the discontinuity can be classified as …

Answer 23

jump discontinuity

Question 24

What are the steps to simplify the limit with complex functions?

Answer 24

1- Simplify the side of the function with complex fractions by multiplying by common factors
2- Re-write original equation to cancel any holes and simplify

Question 25

What are the 3 main restrictions in a function's domain?

Answer 25

1- Denominators CAN NOT EQUAL ZERO 2- All even roots have a domain GREATER or EQUAL to 0 3- All logarithms (inverse of exponential) have a domain GREATER than 0

Question 26

Limit concept allows us to define _____________ ROC in terms of _______ ROC.

Answer 26

instantaneous | average

Question 27

Because average ROC divides change in one variable by change in other, average ROC is _________ at a point where change in the independent variable would be ____

Answer 27

undefined | zero

Question 28

A limit can be expressed in what three ways?

Answer 28

Graphically Numerically Analytically

Question 29

What are 3 ways that a limit might not exist at particular values of x?

Answer 29

Unbounded Oscillating near this value Limit from left side does not equate to that from the right side

Question 30

lim 1/x^2 = | x->0

Answer 30

Infinity

Question 31

lim lxl / x= | x->0

Answer 31

does not exist

Question 32

``` lim sin(1/x) = x->0 ```

Answer 32

does not exist

Question 33

lim 1/x = | x->0

Answer 33

does not exist

Question 34

What are the three ways to algebraically manipulate a limit?

Answer 34

1- cancelling out COMMON FACTORS of rational functions 2- multiplying by expression involving CONJUGATE of a sum or difference to simplify functions involving RADICALS 3- using ALTERNATE forms of TRIG functions

Question 35

A function is _________ on an interval if function is continuous at ____ point in the interval

Answer 35

continous | each

Question 36

In order for a _________-defined function to be continuous at a boundary to the partition of its domain, the value of the expression defining the function on on side must _____ the value on the other side as well as the _____ of the function at the boundary.

Answer 36

piecewise equal value

Question 37

It is possible to remove a discontinuity by ________ the value of the function at that point so it _____ the value of the _____ of the function as x approaches that point.

Answer 37

defining equals limit

Question 38

A vertical asymptote that approaches positive infinity can be identified if ....

Answer 38

The change in the numerator is much greater than the change in the denominator in the positive direction

Question 39

A vertical asymptote that approaches negative infinity can be identified if ....

Answer 39

The change in the numerator is much greater than the change in the denominator in the negative direction

Question 40

Limits of horizontal asymptotes at infinity describes ...

Answer 40

end behavior

Question 41

Horizontal asymptote is the __________ line a function __________ as it heads towards ________ (positive or negative)

Answer 41

horizontal approaches infinity

Question 42

If the denominator grows faster than the numerator in a function, then the function will equal ...

Answer 42

0 or horizontal asympote

Question 43

If the denominator and numerator grows at about the same rate, then the function will equal ....

Answer 43

1

Question 44

If the numerator grows faster than the denominator, then the function will equal...

Answer 44

infinity

Question 45

Rank in order of least to greatest growth. LOGS EXPONENTIALS POLYNOMIALS

Answer 45

Logs Polynomials Exponentials

Question 46

When a limit evaluates a function at infinity,

Answer 46

you can use direct substitution to evaluate

Question 47

If f is a __________ function from a to b, then _____ value between f(a) and f(b) _____ at some point in the interval [a,b].

Answer 47

continuous every exist

Question 48

lim x/sin(x) = | x->0

Answer 48

1

Question 49

lim x/(1 - cos(x)) = | x->0

Answer 49

0