Limits and Continuity

Question 1

What is the general equation of a limit?

Answer 1

lim x->c f(x) = L

Question 2

What does a limit represent?

Answer 2

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.

Question 3

What is a left-handed limit?

Answer 3

The limit of a function as x approaches c from the left. x->c^-

Question 4

What is a right-handed limit?

Answer 4

The limit of a function as x approaches c from the right. x->c^+

Question 5

What should you know about abbreviations?

Answer 5

Do NOT use abbreviations. AP exams will not accept them, so don’t get into the habit of using them.

Question 6

What is the requirement for a limit to exist?

Answer 6

The left-handed and right-handed limits must both exist and be equal.

Question 7

What should you remember about the limits of quotients?

Answer 7

lim x->c f(x)/g(x) = lim x->c f(x) / lim x-> c g(x), provided lim x-c g(x) != 0

Question 8

Average rate of change equation

Answer 8

delta y / delta x = (f(b) - f(a)) / b - a, b != a

Question 9

Difference quotient

Answer 9

(f(x+h) - f(x)) / h, h != 0

Question 10

What do you do with 0 in limits?

Answer 10

0/N = 0
N/0 = does not exist (may be +-infinity, just doesn’t exist in a special way)
0/0 = DO MORE WORK

Question 11

Three conditions for continuity

Answer 11

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.

Question 12

What are the different types of discontinuities?

Answer 12

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).

Question 13

How do you repair a removable discontinuity?

Answer 13

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

Question 14

what is f(x) = sqrt(N-x^2)

Answer 14

SEMI CIRCLE

Question 15

What is the equation of a semi-circle?

Answer 15

sqrt(N-x^2)

Question 16

How would you answer a question using the intermediate value theorem?

Answer 16

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.

Question 17

How does the squeeze theorem work?

Answer 17

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

Question 18

What are the basic trigonometric limits taught in Limits and Continuity?

Answer 18

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

Question 19

Do infinite limits exist?

Answer 19

They do not exist, but in a special way.

Question 20

How do you determine whether a function is approaching positive or negative infinity?

Answer 20

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.

Question 21

What are entrance and exit behaviour?

Answer 21

the limit of f(x) as x approaches infinity is exit behavior.
the limit of f(x) as x approaches -infinity is entrance behavior.

Question 22

Can asymptotes be crossed?

Answer 22

Vertical asymptotes can never be crossed. Horizontal asymptotes can be crosses (they describe the entrance and exit behaviours)

Question 23

What are the orders of magnitude?

Answer 23

1) exponential. 2) polynomial. 3) logarithmic.

Question 24

How do you find horizontal and vertical asymptotes?

Answer 24

Vertical asymptotes) Right and left hand Limits of the 0s in the denominator (check if they equal infinity)
Horizontal asymptotes) +-infinity limits

Question 25

sqrt(x^2)

Answer 25

abs(x)

Question 26

When do you have to use words to answer something?

Answer 26

When it asks for justification

Question 27

The first thing you should try for any function is…

Answer 27

Direct substitution

Question 28

How do you describe start and end behaviours?

Answer 28

x->infinity y->value
x->-infinity y->value

Question 29

How do you write an intercept?

Answer 29

As a coordinate (x, y)

Question 30

How do you write asymptotes?

Answer 30

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

Question 31

Table format

Answer 31

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

Question 32

How do you define a function in the Ti-nspire calculator?

Answer 32

Define f1(x)=EQUATION

Question 33

When can you truncate zeroes?

Answer 33

When there are no values after it. Ex, 4/2=2, .001/3=.000 (0.000333…)

Question 34

What should you check when using a table?

Answer 34

If an asymptote is approached or the value is not approaching a specific value

Question 35

If f(x) is not approaching a value when using a table, what do you write in the box

Answer 35

f(x) does not approach a specific value

Question 36

Does L depend on c?

Answer 36

No

Question 37

Can a point have more than one limit?

Answer 37

No.

Question 38

What do you need to remember about radical and logarithmic functions?

Answer 38

Even degree radical functions domains are [0,infinity). Logarithmic functions domains are (0,infinity). This assumes no horizontal shift

Question 39

How do you write the secant between two points?

Answer 39

Msec (sec is subscript) = (y2-y1) / (x2-x1)

Question 40

How do you find a secant line that intersects the two points?

Answer 40

Remember y=A(x-k)+h
Where A is the slope and k and h are translations

Question 41

Intermediate value theorem, what must you say first

Answer 41

Since f(x) is continuous over the interval (interval)