Limits

Question 1

Standard Limit function

Answer 1

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 2

Right-hand limit

Answer 2

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 3

Left-hand limit

Answer 3

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 4

Limit exists if…

Answer 4

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

Question 5

Discontinuity Types

Answer 5

1) Hole
2) Jump
3) Piecewise

Question 6

Hole Discontinuity

Answer 6

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

Question 7

Jump Discontinuity

Answer 7

The curve continues at another point on a new trend

Question 8

Limit laws

Answer 8

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

Question 9

Squeeze theorem

Answer 9

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 10

Limits of rational functions

Answer 10

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

Question 11

Infinate limits

Answer 11

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

Question 12

Vertical Asymptote

Answer 12

Input at which the limit is ±∞

Question 13

Horizontal Assymptote

Answer 13

The limit as x→±∞

Question 14

Continuous if…

Answer 14

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 15

Continuity of polynomial functions

Answer 15

Always continuous

Question 16

Continuity of rational functions

Answer 16

p(x)/q(x)

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

Question 17

Continuity of composite functions

Answer 17

As long as f is continuous and g is continuous

Then (f o g) is also continuous

Question 18

Left continuous

Answer 18

Lim x→a- f(x)=f(a)

Question 19

Right continuous

Answer 19

Lim x→a+ f(x)=f(a)

Question 20

Continuity on an interval

Answer 20

The function is continuous at all points INCLUDED within that interval

Question 21

Continuity of roots or powers

Answer 21

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

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

Question 22

Continuity of inverse functions

Answer 22

If the original function f(x) is continuous

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

Question 23

Common continuous transcendental functions

Answer 23

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

Question 24

Intermediate value theorem

Answer 24

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

Question 25

Proofing Limits

Answer 25

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

Remember 0

Question 26

δ of Limits

Answer 26

The maximum value of |x-a|. Often ∞.

Remember 0

Question 27

ε of limits

Answer 27

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

Remember 0

Question 28

Speed

Answer 28

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

Question 29

Average velocity

Answer 29

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

s is the position function in respect time(t)