Limit of A Function
Suppose f(x) is a real-valued function and c is a real number. The expression
means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, the above equation can be read as "the limit of f of x, as x approaches c, is L".
Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass, formalized the definition of the limit of a function as the above definition, which became known as the (ε, δ)-definition of limit in the 19th century. The definition uses ε (the lowercase Greek letter epsilon) to represent a small positive number, so that "f(x) becomes arbitrarily close to L" means that f(x) eventually lies in the interval (L - ε, L + ε), which can also be written using the absolute value sign as |f(x) - L| < ε. The phrase "as x approaches c" then indicates that we refer to values of x whose distance from c is less than some positive number δ (the lower case Greek letter delta)—that is, values of x within either (c - δ, c) or (c, c + δ), which can be expressed with 0 < |x - c| < δ. The first inequality means that the distance between x and c is greater than 0 and that x ≠ c, while the second indicates that x is within distance δ of c.
Note that the above definition of a limit is true even if f(c) ≠ L. Indeed, the function f(x) need not even be defined at c.
For example, if
then f(1) is not defined (see division by zero), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2:
| f(0.9) | f(0.99) | f(0.999) | f(1.0) | f(1.001) | f(1.01) | f(1.1) |
| 1.900 | 1.990 | 1.999 | ⇒ undefined ⇐ | 2.001 | 2.010 | 2.100 |
Thus, f(x) can be made arbitrarily close to the limit of 2 just by making x sufficiently close to 1.
In other words,
In addition to limits at finite values, functions can also have limits at infinity. For example, consider
- f(100) = 1.9900
- f(1000) = 1.9990
- f(10000) = 1.99990
As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish just by picking x sufficiently large. In this case, the limit of f(x) as x approaches infinity is 2. In mathematical notation,
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Famous quotes containing the words limit of, limit and/or function:
“Can you find out the deep things of God? Can you find out the limit of the Almighty?”
—Bible: Hebrew, Job 11:7.
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“Uses are always much broader than functions, and usually far less contentious. The word function carries overtones of purpose and propriety, of concern with why something was developed rather than with how it has actually been found useful. The function of automobiles is to transport people and objects, but they are used for a variety of other purposes—as homes, offices, bedrooms, henhouses, jetties, breakwaters, even offensive weapons.”
—Frank Smith (b. 1928)