Limits and Continuity Flashcards
What is the general equation of a limit?
lim x->c f(x) = L
What does a limit represent?
The value f(x) can be made as close as we please to L, for x sufficiently close to c, but not equal to c.
What is a left-handed limit?
The limit of a function as x approaches c from the left. x->c^-
What is a right-handed limit?
The limit of a function as x approaches c from the right. x->c^+
What should you know about abbreviations?
Do NOT use abbreviations. AP exams will not accept them, so don’t get into the habit of using them.
What is the requirement for a limit to exist?
The left-handed and right-handed limits must both exist and be equal.
What should you remember about the limits of quotients?
lim x->c f(x)/g(x) = lim x->c f(x) / lim x-> c g(x), provided lim x-c g(x) != 0
Average rate of change equation
delta y / delta x = (f(b) - f(a)) / b - a, b != a
Difference quotient
(f(x+h) - f(x)) / h, h != 0
What do you do with 0 in limits?
0/N = 0
N/0 = does not exist (may be +-infinity, just doesn’t exist in a special way)
0/0 = DO MORE WORK
Three conditions for continuity
f(c) is defined (c is in the domain of f)
lim x->c f(x) exists
lim x->c f(x) = f(c)
Check these conditions in this order. Soon as one condition is not met, stop, and use it as reasoning for why f is not continuos at c.
What are the different types of discontinuities?
Removeable discontinuity: Limit at c exists, but f(c) does not exist or has a different value. This can be ‘fixed’ making it continuous.
Jump discontinuity: Left and right handed limits exist but are not equal to each other.
Infinite discontinuity: Left or right handed limit or both are infinite.
Oscillating discontinuity: Neither the left nor right-hand limit exists (it does not settle on a specific value).
How do you repair a removable discontinuity?
Determine the point of discontinuity. Redefine f(x) with a new piecewise function that includes the original function if x does not equal to point of discontinuity, and the value that x would be if x equals the point of discontinuity.
ex: f(x) = x(x-1)/x
x != 0
f(x) = {x(x-1)/x, x != 0, -1, x = 0
what is f(x) = sqrt(N-x^2)
SEMI CIRCLE
What is the equation of a semi-circle?
sqrt(N-x^2)
How would you answer a question using the intermediate value theorem?
Ex
Since f(x) is a polynomial, it is continuous on the interval [-2,0], since f(-2)=4 > 0, and f(0)=-6 < 0, by the intermediate value theorem, there is a 0 between -2 and 0.
How does the squeeze theorem work?
If f(x)<=g(x)<=h(x) over the interval (interval containing c)
and lim x->c f(x) = limt x->x h(x) = L
Then lim x->c g(x) = L
What are the basic trigonometric limits taught in Limits and Continuity?
Works with substitution
lim x->0 sin(x) = 0
lim x-> 0 cos(x) = 1
lim x->c sin(x) = sin(c)
lim x->c cos(x) = cos(c)
Don’t work with substitution
lim x->0 sin(x)/x = 1
lim x->0 (cos(x)-1) / x = 0
Do infinite limits exist?
They do not exist, but in a special way.
How do you determine whether a function is approaching positive or negative infinity?
First, try DIRECT SUBSTITUTION, this may reveal 0/0 which needs more work.
Determine how the numerator and denominator approach their values.
Does the numerator approach from positive or negative? What about the denominator?
-/+ or +/- = -infinity
+/+ or -/- = +infinity
Do this for both the right and left handed limits, if they are equal, the limit exists, if not, the limit does not exist.
What are entrance and exit behaviour?
the limit of f(x) as x approaches infinity is exit behavior.
the limit of f(x) as x approaches -infinity is entrance behavior.
Can asymptotes be crossed?
Vertical asymptotes can never be crossed. Horizontal asymptotes can be crosses (they describe the entrance and exit behaviours)
What are the orders of magnitude?
1) exponential. 2) polynomial. 3) logarithmic.
How do you find horizontal and vertical asymptotes?
Vertical asymptotes) Right and left hand Limits of the 0s in the denominator (check if they equal infinity)
Horizontal asymptotes) +-infinity limits
sqrt(x^2)
abs(x)
When do you have to use words to answer something?
When it asks for justification
The first thing you should try for any function is…
Direct substitution
How do you describe start and end behaviours?
x->infinity y->value
x->-infinity y->value
How do you write an intercept?
As a coordinate (x, y)
How do you write asymptotes?
Vertical asymptotes are written are left and right hand limits.
Horizontal limits like x->+-infinity y->value but if entrance and exits agree, y=value can be used
Table format
Top left, x
Bottom left, f(x)=equation
The value that x is approaching is in the middle with arrows on either side pointing to it, underneath: f(x) approaches VALUE
How do you define a function in the Ti-nspire calculator?
Define f1(x)=EQUATION
When can you truncate zeroes?
When there are no values after it. Ex, 4/2=2, .001/3=.000 (0.000333…)
What should you check when using a table?
If an asymptote is approached or the value is not approaching a specific value
If f(x) is not approaching a value when using a table, what do you write in the box
f(x) does not approach a specific value
Does L depend on c?
No
Can a point have more than one limit?
No.
What do you need to remember about radical and logarithmic functions?
Even degree radical functions domains are [0,infinity). Logarithmic functions domains are (0,infinity). This assumes no horizontal shift
How do you write the secant between two points?
Msec (sec is subscript) = (y2-y1) / (x2-x1)
How do you find a secant line that intersects the two points?
Remember y=A(x-k)+h
Where A is the slope and k and h are translations
Intermediate value theorem, what must you say first
Since f(x) is continuous over the interval (interval)
