## Chapter 4 - Functions

MM - Page 13 of 15
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## Limit Functions

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 Functions - Limit Functions - Contents

A limit function is used to examine other functions and has the following form: the limit of f(x) as x approaches n. A limit function can look at other functions f(x) as the independent variable x gets closer to a number n and provides the results as a limit which can be a number, zero, infinity or it can be undefined. For a limit to exist at x = n the function must be defined at that point. The limit of a function can be used to prove the function's continuity at that point. If the function is convergent it approaches a constant number, then the limit exists and the function may be considered continuous. A convergent limit means the summation function series for that function may have a solution at that point. The limit function is also used to create the special number e used in the exponential function and natural logarithmic functions. The domain and the range of a limit function are defined by the equation being tested.

 Functions - Limit Functions - Examples

Limit function:
If
y = 2x + 1,
y
equals the limit as x approaches infinity
of the function two times
x plus one,
y
= 1, 3, 5, 7, 9, 11, 13 . . . ,
so
y also approaches infinity as x gets larger.

Limit function:
If
y = 1/(x + 1),
y
equals the limit as x approaches infinity
of the function one divided by the quantity
x plus one,
y
= 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7 . . . ,
here
y approaches zero as x gets larger approaching infinity,
so the limit is convergent.

 Functions - Limit Functions - Mysteries

Math Mysteries: The special number e is created using the limit function e = (1 + 1/ x) to the power of x. This unique equation causes the limit to approach an irrational number that has mysterious properties as the limit of x approaches infinity. As x increases incrementally from one, the limit climbs 2, 2.25, 2.37, 2.44, 2.49, etc., up to e equal to 2.2718281828459045 . . . which is an infinitely long nonrepeating decimal.

 Functions - Sections - Chapters 1 - Function Definition 2 - Function Notation 3 - Function Domain 4 - Function Range 5 - Composite Functions 6 - Inverse Functions 7 - Linear Functions 8 - Power Functions 9 - Quadratic Functions 10 - Logarithmic Functions 11 - Exponential Functions 12 - Factorial Functions 13 - Limit Functions 14 - Summation Functions 15 - Percentage Functions

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