Limits Flashcards

1
Q

Instantaneous Velocity

A
Limit as (t-ti)→0 for average velocity curve 
As long as (t+ti)=[moment in question]
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2
Q

Standard Limit function

A

Lim x→a f(x)=L

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

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3
Q

Right-hand limit

A

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’

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4
Q

Left-hand limit

A

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’

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5
Q

Limit exists if…

A

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

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6
Q

Discontinuity Types

A

1) Hole
2) Jump
3) Piecewise

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7
Q

Hole Discontinuity

A

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

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8
Q

Jump Discontinuity

A

The curve continues at another point on a new trend

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9
Q

Limit laws

A

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

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10
Q

Squeeze theorem

A

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

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11
Q

Limits of rational functions

A

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

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12
Q

Infinate limits

A

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

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13
Q

Vertical Asymptote

A

Input at which the limit is ±∞

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14
Q

Horizontal Assymptote

A

The limit as x→±∞

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15
Q

Continuous if…

A

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

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16
Q

Continuity of polynomial functions

A

Always continuous

17
Q

Continuity of rational functions

A

p(x)/q(x)

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

18
Q

Continuity of composite functions

A

As long as f is continuous and g is continuous

Then (f o g) is also continuous

19
Q

Left continuous

A

Lim x→a- f(x)=f(a)

20
Q

Right continuous

A

Lim x→a+ f(x)=f(a)

21
Q

Continuity on an interval

A

The function is continuous at all points INCLUDED within that interval

22
Q

Continuity of roots or powers

A

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

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

23
Q

Continuity of inverse functions

A

If the original function f(x) is continuous

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

24
Q

Common continuous transcendental functions

A

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

25
Q

Intermediate value theorem

A

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

26
Q

Proofing Limits

A

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

Remember 0< |x-a|

27
Q

δ of Limits

A

The maximum value of |x-a|. Often ∞.

Remember 0< |x-a|

28
Q

ε of limits

A

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

Remember 0< |f(x)-L|

29
Q

Speed

A

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

30
Q

Average velocity

A

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

s is the position function in respect time(t)