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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