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