Flashcards in Limits Deck (30):
Limit as (t-ti)→0 for average velocity curve
As long as (t+ti)=[moment in question]
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'
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'
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'
Limit exists if...
Lim x→a- f(x)=Lim x→a+ f(x)=L, right and left hand limit are equal
One point on the curve has its own output, unique from the trend of the other points
The curve continues at another point on a new trend
Basically, what happens to a function, also happens to the limit of that function (or functions)
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
Limits of rational functions
Factor the denominator out of the numerator and solve the limit from there
The output of the function continues to grow as the input approaches a value, but does simply keeps growing
Input at which the limit is ±∞
The limit as x→±∞
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
Continuity of polynomial functions
Continuity of rational functions
Continuous at all points for which q(x)≠0
Continuity of composite functions
As long as f is continuous and g is continuous
Then (f o g) is also continuous
Lim x→a- f(x)=f(a)
Lim x→a+ f(x)=f(a)
Continuity on an interval
The function is continuous at all points INCLUDED within that interval
Continuity of roots or powers
As long as the original function f(x) is continuous
Then [f(x)]^(m/n) is continuous too
Continuity of inverse functions
If the original function f(x) is continuous
Then f^-1(x) is continuous on those same intervals
Common continuous transcendental functions
All trig functions, all inverse trig functions, exponential functions (b^x), logarithmic functions
Intermediate value theorem
On any continuous interval [a,b], there is an input for every output between 'a' and 'b'
1) Find δ of limit. The maximum value of |x-a|. Often ∞.
δ of Limits
The maximum value of |x-a|. Often ∞.
ε of limits
The maximum value of |f(x)-L|
Absolute value of velocity