Unit 1 - Limits & Continuity Flashcards
What is the secant line?
Straight line that connects two points on a function; equivalent to the AVERAGE ROC
What is the tangent line?
Straight line that connects two very close points or one point on a function; equivalent to the INSTANTANEOUS ROC
How do you read a limit?
“The limit x approaches (x-value) of f(x) is …”
T or F: Limit and y-value could be the same value but does not have to be the same value.
True
What information does a limit give you?
A limit tells you what point a function is APPROACHING , but it doesn’t tell you what the VALUE of that particular point is.
What is a one-sided limit?
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
How do you notate a limit from the left side?
Add a NEGATIVE sign after the x-value
How do you notate a limit from the right side?
Add a POSITIVE sign after the x-value
How can you determine what value a limit is approaching from the table?
As the x-values get increasingly closer to the point of interest, the y-value will also increasingly approach a value.
How do you evaluate a limit at a point?
direct substitution or factor and cancel
If two sides of a limit does not meet, then ….
The limit DOES NOT exist
lim sin(x)/x = x->0
1
lim (1 - cos(x))/x =
x->0
0
If you completely factor the numerator and denominator of an expression and no factors cancel, then…
the limit DOES NOT exist
When an indeterminate form appears, what strategies can you use to determine the limit
Complex fractions
Rationalizing with Radicals
Squeeze Theorem
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->aHow do you solve absolute value limits?
Find numbers that are close to x to substitute as x
Three conditions of continuity
1- f(c) is defined (with c=domain)
2- lim f(x) exists
x->c
3- lim f(x) = f(c)
x->cThree types of discontinuities
Hole/ Removable
Vertical Asymptote/ Non-removable
Jump/ Non-removable
In a function, discontinuities are found when …
the denominator is equivalent to 0
If the factors in an expression cancels, then the discontinuity can be classified as…
hole discontinuity
If the factors in an expression does not cancels, then the discontinuity can be classified as…
an vertical asymptote
If the left side of a limit is different from the right side of a limit, then the discontinuity can be classified as …
jump discontinuity
What are the steps to simplify the limit with complex functions?
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
What are the 3 main restrictions in a function’s domain?
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
Limit concept allows us to define _____________ ROC in terms of _______ ROC.
instantaneous
average
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 ____
undefined
zero
A limit can be expressed in what three ways?
Graphically
Numerically
Analytically
What are 3 ways that a limit might not exist at particular values of x?
Unbounded
Oscillating near this value
Limit from left side does not equate to that from the right side
lim 1/x^2 =
x->0
Infinity
lim lxl / x=
x->0
does not exist
lim sin(1/x) = x->0
does not exist
lim 1/x =
x->0
does not exist
What are the three ways to algebraically manipulate a limit?
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
A function is _________ on an interval if function is continuous at ____ point in the interval
continous
each
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.
piecewise
equal
value
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.
defining
equals
limit
A vertical asymptote that approaches positive infinity can be identified if ….
The change in the numerator is much greater than the change in the denominator in the positive direction
A vertical asymptote that approaches negative infinity can be identified if ….
The change in the numerator is much greater than the change in the denominator in the negative direction
Limits of horizontal asymptotes at infinity describes …
end behavior
Horizontal asymptote is the __________ line a function __________ as it heads towards ________ (positive or negative)
horizontal
approaches
infinity
If the denominator grows faster than the numerator in a function, then the function will equal …
0 or horizontal asympote
If the denominator and numerator grows at about the same rate, then the function will equal ….
1
If the numerator grows faster than the denominator, then the function will equal…
infinity
Rank in order of least to greatest growth.
LOGS
EXPONENTIALS
POLYNOMIALS
Logs
Polynomials
Exponentials
When a limit evaluates a function at infinity,
you can use direct substitution to evaluate
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].
continuous
every
exist
lim x/sin(x) =
x->0
1
lim x/(1 - cos(x)) =
x->0
0
