Limit of a Function
The limit of a function is a fundamental concept in calculus that describes how a function behaves as it approaches a particular point. Essentially, it captures the idea that one can approach a specific value by making increasingly smaller increments in their input values. For continuous functions, this approach is straightforward, as the limit can be determined by simply evaluating the function at that point. However, for functions that are not defined at certain points, limits provide a way to analyze their behavior around those undefined points.
For example, in the case of a function that approaches a value from either direction, the limit can be expressed as a two-sided limit, which gives the same result when approached from the left or right. In contrast, if a function is discontinuous, the limit may exist even when the function itself does not. Historical contributions from mathematicians like l'Hôpital and Cauchy have been crucial in formalizing the rules governing limits, which are essential for understanding both differential and integral calculus. This exploration of limits not only addresses theoretical challenges but also has practical applications in various fields, including physics and engineering.
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Limit of a Function
Ancient thinkers had major conceptual challenges with the concepts of zero and infinity, tied up with abstract notions of nothingness, nonexistence, and unattainable perfection. As a classic example by Zeno of Elea (c. 490–420 BCE), consider an ant moving a certain distance. First, the ant must cover half the distance, then half the remaining distance, then half the next remaining distance, and so forth. The ant must cover an infinite number of half-distances, each of which takes some finite amount of time to cross. An infinite number of finite times seemed as if it would be infinite, so it will take the ant an infinite amount of time to cover the full distance.
Even the proto-scientific thinking available in the ancient Greek world made it clear that this contradicted the clear evidence that ants could cross distances, but mathematicians were hard pressed how to address Zeno’s paradox because of the philosophical complications caused by zero and infinity.
Solving problems such as this, and developing the mathematics of calculus, would require talking about the unattainable concept of infinity through the method of approaching a limit.
Overview
The limit of a function is a simple concept, though the language to define it precisely in mathematical terms can seem intimidating. For a continuous function with no gaps or breaks, the concept is relatively straightforward: the limit of a function represents, generally speaking, the idea that it is possible to approach a given point on the function by taking ever more tiny steps to approach that point.
Consider the number sequence of halves from Zeno’s paradox.
As the number of half-distances, n, gets larger, the remaining distance gets smaller. It is helpful to look at the graph (Figure 1) representing the distance left for the nth term in the series.
As the number n gets larger, as it approaches infinity, the term gets smaller and smaller, approaching ever closer to the value of 0. Mathematicians would then say that "the limit of one over 2n as n approaches infinity equals 0," expressed in mathematical notation as:
This one-sided limit approaches the limit from only one direction as n increases. There is no other way for the value of n to approach infinity.
A two-sided limit can also exist, such as
for f(x) = −x2 + 3 (Figure 2). This limit can be approached from either the left or right directions. Because the function is continuous, this limit from each direction yields the same value, though this isn’t the case for non-continuous functions.
Finding the limit of f(x) at 0 can also be done by algebraically evaluating f(0) = 3 or looking at the graph. Some functions cannot be defined at a point, such as the following function g(x), and these cases make the use of limits more clear.
Since the point g(0) is undefined, this function doesn’t exist at that point, but the limit of the function does exist there.
Mathematicians such as Guillaume François l’Hopital (1661–1715 CE) and Augustin-Louis Cauchy (1789–1857 CE) contributed heavily to the understanding of rules about how to work with limits, largely due to the central role they play in differential and integral calculus.
Bibliography
Borovik, Alexandre, and Mikhail Katz. "Who Gave You the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus." Foundations of Science 17.3 (2012): 245-76.
Hornsby, John E., Margaret L. Lial, Gary K. Rockswold. A Graphical Approach to Precalculus with Limits. 6th ed. London: Pearson, 2014.
Larson, Ron. Precalculus with Limits. Boston: Cengage, 2014.
Ouellette, Jennifer. The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse. New York: Penguin, 2010.
Pourciau, Bruce. "Newton and the Notion of Limit." Historia Mathematica 28.1 (2001): 18-30.
Schuette, Paul. "A Question of Limits." Mathematics Magazine 77.1 (2004): 61-68.