Limits

Question 1

Instantaneous Velocity

Answer 1

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

Question 2

Standard Limit function

Answer 2

Lim x→a f(x)=L

As the input gets closer and closer to ‘a’ the output of the function comes closer to ‘L’

Question 3

Right-hand limit

Answer 3

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’

Question 4

Left-hand limit

Answer 4

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’

Question 5

Limit exists if…

Answer 5

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

Question 6

Discontinuity Types

Answer 6

1) Hole
2) Jump
3) Piecewise

Question 7

Hole Discontinuity

Answer 7

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

Question 8

Jump Discontinuity

Answer 8

The curve continues at another point on a new trend

Question 9

Limit laws

Answer 9

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

Question 10

Squeeze theorem

Answer 10

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

Question 11

Limits of rational functions

Answer 11

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

Question 12

Infinate limits

Answer 12

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

Question 13

Vertical Asymptote

Answer 13

Input at which the limit is ±∞

Question 14

Horizontal Assymptote

Answer 14

The limit as x→±∞

Question 15

Continuous if…

Answer 15

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

Question 16

Continuity of polynomial functions

Answer 16

Always continuous

Question 17

Continuity of rational functions

Answer 17

p(x)/q(x)

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

Question 18

Continuity of composite functions

Answer 18

As long as f is continuous and g is continuous

Then (f o g) is also continuous

Question 19

Left continuous

Answer 19

Lim x→a- f(x)=f(a)

Question 20

Right continuous

Answer 20

Lim x→a+ f(x)=f(a)

Question 21

Continuity on an interval

Answer 21

The function is continuous at all points INCLUDED within that interval

Question 22

Continuity of roots or powers

Answer 22

As long as the original function f(x) is continuous

Then [f(x)]^(m/n) is continuous too

Question 23

Continuity of inverse functions

Answer 23

If the original function f(x) is continuous

Then f^-1(x) is continuous on those same intervals

Question 24

Common continuous transcendental functions

Answer 24

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

Question 25

Intermediate value theorem

Answer 25

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

Question 26

Proofing Limits

Answer 26

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

Remember 0< |x-a|

Question 27

δ of Limits

Answer 27

The maximum value of |x-a|. Often ∞.

Remember 0< |x-a|

Question 28

ε of limits

Answer 28

The maximum value of |f(x)-L|

Remember 0< |f(x)-L|

Question 29

Speed

Answer 29

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

Question 30

Average velocity

Answer 30

v=(s(t)-s(ti))/(t-ti)

s is the position function in respect time(t)