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Flashcards in Limits Deck (30):
1

Instantaneous Velocity

Limit as (t-ti)→0 for average velocity curve
As long as (t+ti)=[moment in question]

2

Standard Limit function

Lim x→a f(x)=L
As the input gets closer and closer to 'a' the output of the function comes closer to 'L'

3

Right-hand limit

Lim x→a+ f(x)=L
As the input gets closer and closer to 'a' from the positive direction the output of the function comes closer to 'L'

4

Left-hand limit

Lim x→a- f(x)=L
As the input gets closer and closer to 'a' from the negative direction the output of the function comes closer to 'L'

5

Limit exists if...

Lim x→a- f(x)=Lim x→a+ f(x)=L, right and left hand limit are equal

6

Discontinuity Types

1) Hole
2) Jump
3) Piecewise

7

Hole Discontinuity

One point on the curve has its own output, unique from the trend of the other points

8

Jump Discontinuity

The curve continues at another point on a new trend

9

Limit laws

Basically, what happens to a function, also happens to the limit of that function (or functions)

10

Squeeze theorem

If a function greater and a function less than the desired function are equal to a given value, then the desired function must be equal to that value

11

Limits of rational functions

Factor the denominator out of the numerator and solve the limit from there

12

Infinate limits

The output of the function continues to grow as the input approaches a value, but does simply keeps growing

13

Vertical Asymptote

Input at which the limit is ±∞

14

Horizontal Assymptote

The limit as x→±∞

15

Continuous if...

1) Point 'a' is within the domain
2) Lim x→a- f(x)=Lim x→a+ f(x)=L, right and left hand limit are equal
3) Lim x→a f(x)=f(a), limit is always equal to the output at that point

16

Continuity of polynomial functions

Always continuous

17

Continuity of rational functions

p(x)/q(x)
Continuous at all points for which q(x)≠0

18

Continuity of composite functions

As long as f is continuous and g is continuous
Then (f o g) is also continuous

19

Left continuous

Lim x→a- f(x)=f(a)

20

Right continuous

Lim x→a+ f(x)=f(a)

21

Continuity on an interval

The function is continuous at all points INCLUDED within that interval

22

Continuity of roots or powers

As long as the original function f(x) is continuous
Then [f(x)]^(m/n) is continuous too

23

Continuity of inverse functions

If the original function f(x) is continuous
Then f^-1(x) is continuous on those same intervals

24

Common continuous transcendental functions

All trig functions, all inverse trig functions, exponential functions (b^x), logarithmic functions

25

Intermediate value theorem

On any continuous interval [a,b], there is an input for every output between 'a' and 'b'

26

Proofing Limits

1) Find δ of limit. The maximum value of |x-a|. Often ∞.
Remember 0< |x-a|

27

δ of Limits

The maximum value of |x-a|. Often ∞.
Remember 0< |x-a|

28

ε of limits

The maximum value of |f(x)-L|
Remember 0< |f(x)-L|

29

Speed

Absolute value of velocity
|(p-pi)/(t-ti)|
p- position
t- time

30

Average velocity

v=(s(t)-s(ti))/(t-ti)
s is the position function in respect time(t)