In "well-behaved" functions (i.e. in continuous ones), the limit, as x approaches c, can be found by directly substituting c for x. For example, if f(x) = 7; as x approaches 32, the limit is 7 (the limit of a constant is a constant). Another example is f(x) = 2x - 5; in that situation, as x approaches 3, f(x) approaches f(3) = 2·3 - 5 = 1.
Limits are more interesting when they are unreachable; for example: if f(x) = (x³ - 1) / (x - 1) then, x cannot be set equal 1 (as that would result in division by zero). To say that more formally the value 1 is not in the function domain.
Here, however, f(x) does approach some number c, as x approaches 1. One can compute f(0.9) = 2.71, f(0.99) = 2.9701, f(0.999) = 2.997001, f(1.1) = 3.31, f(1.01) = 3.0301, f(1.001) = 3.003001.
We see that, as x approaches 1, f(x) approaches 3; but x is never equal to 1 and f(x) never equals 3. The limit can be verified using algebra; since: (x2 + x + 1)(x - 1) = (x3 - 1)...if g(x) = x2 + x + 1, g(x) = f(x) (for any x ≠ 1). g(x) and f(x) are identical at every point, except 1; f(x) has a 'hole' in its domain at 1. Via direct substitution (x = 1) one can determine the limit of g(x) to be 12 + 1 + 1 = 3; which is the value f(x) approaches, as it approaches the hole.
Here we have used the following rule:
if it so happens that if g(x) = f(x) for all values of x, except for c; then, so long as g(x) has a limit, that limit will be equal to the limit of f(x).
The above also shows that whether or not f(c) exists has no bearing on whether or not the limit of f(x) (as x approaches c) exists. If f(c) exists, then its value has also no bearing on the limit.
Not every function has a limit at every point. Consider:
When trying to evaluation a limit by simply substitution c into the function, one often would have to divide by zero, which is of course impossible. Here l'Hôpital's rule helps: If f(c)=0, g(c)=0, and both derivatives f'(c) and g'(c) are defined, then:
The "squeeze theorem" (or "pinching theorem" as it is also known) is a theorem telling us that if three functions f(x), g(x) and h(x) are given such that h(x) ≤ f(x) ≤ g(x) and if the limit, L, (as x approaches c) of h(x) is equal to the limit (as x approaches c) of g(x); the limit of f(x) not only exists, but is also equal to L. The function f(x) is "squeezed" between g(x) and h(x).
One might ask whether there is some relationship between |f(x) - L| (the absolute value [of the function, at x, minus its limit, L, as x approaches c]) and |x - c| (the absolute value [of x minus the number it is approaching]). For every number, ε > 0, there is some number, δ > 0; such that, if 0 < |x - c| < δ; then, |f(x) - L| < ε. In other words, if the distance between x and c is less than δ; then, the distance between f(x) and L is less than ε.
For example; the limit of (3x - 2), as x approaches 3, is 7; f(3) = 7. If one determines that the absolute value [of the function, at x, minus its limit (as x approaches 3)], should be less than 0.003 (that is, one is attempting to determine what value of x will generate an f(x) within 0.003 of the limit of f(x), as x approaches c = 3); then, one can write: |(3x - 2) - 7| = |3x - 9| = 3|x - 3| < 0.003. Noting that 0 < |x - c| < δ and that, in this situation, c = 3; we can write 0 < |x - 3| < δ and since 3|x - 3| < 0.003; it is only logical to conclude that |x - 3| < 0.003 / 3 = 0.001; and thus, any value within 0.001, of 3, will produce a value within 0.003 of f(x)'s limit (as x approaches 3); that is, a value within 0.003 of 7. For instance, f(3.001)=7.003.Case of continuous functions
Limits not directly accessible
Limits do not always exist
L'Hôpital's rule, and division by zero
Three notable limits
The "Squeeze theorem"
Approaching a limit