Calculus 2 Notes Limits
Here is a summary of the important concepts in Calculus 2, with a focus on limits:
1. Definition of a limit: A limit of a function f(x) at a point x = a is a value L such that, as x
approaches a, the values of f(x) get arbitrarily close to L. The limit is denoted by the
symbol "lim".
2. One-sided limits: A limit of a function f(x) at a point x = a can be either one-sided or twosided. A one-sided limit is a limit where the function is only approaching the limit from
one direction, either from the left or from the right.
3. Infinite limits: A limit of a function f(x) at a point x = a is said to be infinite if the values of
f(x) become arbitrarily large as x approaches a.
4. Determining limits: Limits can be determined using algebraic manipulations, by
considering the graph of the function, or by using L'Hopital's rule.
5. Continuous functions: A function is said to be continuous at a point x = a if its limit at
that point exists and is equal to the function's value at that point. A function is said to be
continuous over an interval if it is continuous at every point in that interval.
6. Discontinuous functions: A function is said to be discontinuous at a point x = a if its limit
at that point does not exist or is not equal to the function's value at that point.
7. Asymptotes: An asymptote is a line that the graph of a function approaches arbitrarily
closely, but never touches or crosses.
8. Sandwich theorem (Squeeze theorem): The sandwich theorem states that if two
functions f(x) and g(x) satisfy f(x) <= g(x) <= h(x) for all x in an interval, except possibly at
x = a, and if lim f(x) = lim g(x) = lim h(x) = L as x approaches a, then lim g(x) = L as x
approaches a.
9. Epsilon-delta definition of a limit: The epsilon-delta definition of a limit is a formal way
of defining the concept of a limit. It states that for any positive number epsilon, there
exists a positive number delta such that for all x, if 0 < |x - a| < delta, then |f(x) - L| <
epsilon.
These are some of the key concepts in Calculus 2 related to limits. Understanding these
concepts and being able to apply them is crucial to mastering Calculus and related subjects.