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Level 557

Limits & the Definition of the Derivative


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Tangent Line
A line that has the same direction as the curve at the point of contact; the best linear approximation to a curve at a point
Secant Line
a line that intersects a circle at two points on the same plane
the limit of f(x) (as x approaches the number "a") is equal to L if...
...the values of f(x) can be made as close as desired to L (arbitrarily close) by choosing a value of x sufficiently close to a (but ≠a) from either side.
...the limit may exist.
If the form of a limit is 0/0, then....
[a non-zero number] / 0, then...
...the limit is not going to exist.
the limit of f(x) as x approaches the number a from the left is equal to L if...
...the values of f(x) can be made arbitrarily close to L by choosing values of x *less* than the number a that are sufficiently close to the number a.
the limit of f(x) as x approaches the number a is equal to L if...
...for every number ε>0, there is some number δ>0 such that the following holds: if 0<|x - a|<δ, then |f(x) - L|<ε.
The Direct Substitution Property
If f is a *polynomial* or a *rational* function and a is in the domain of f, then the limit (as x approaches a) of f(x) is the value of the function f(a).
A function f is continuous on an interval if...
...it is continuous at every number in the interval.
Theorem: Regarding Operations with Continuous Functions
If f and g are continuous at a and c is a constant, then the functions pictured here are also continuous at x = a.
Theorem: Regarding Types of Functions Known to be Continuous on their Domains
The following types of functions are continuous at every number in their domains:
Theorem: Regarding Composition of Continuous Functions
If g is continuous at the number "a" and f is continuous at the number g(a), then the composite function f(g(x)) is continuous at the number "a".
The Premise of the Intermediate Value Theorem
If f is continuous on the *closed* interval [a,b] and N is any number between f(a) and f(b), where f(a)≠f(b)...
The Conclusion of the Intermediate Value Theorem
...then there exists a number c in (a,b) such that f(c) = N.
Theorem: Regarding Differentiable and Continuous Functions
If f is differentiable at a, then f is continuous at a.