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Limit (calculus)

In calculus, the limit of a function (or sequence) is the value that is approached as the function input approaches some other value.[cite:@wikipedia2025limit][cite:@openstax2020calculus] The limit formalises the notion of a function, graph, or sequence coming closer and closer to some specific value, without necessarily ever reaching it. The limit is at the heart of calculus and is the central mechanism used to define continuity, derivatives, and integrals.

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Complex fraction

Given a limit of a "complex fraction" such as $\lim_{x \to 1} \frac{\frac{1}{x + 1} - \frac{1}{2}}{x - 1},$ you can attempt to solve by multiplying by $\frac{lcd (\frac{1}{x + 1}, \frac{1}{2})}{lcd (\frac{1}{x + 1}, \frac{1}{2})}$ to hopefully get a factorable expression.

Notation

The limit of a function $f$ is usually written as $\lim_{x \to c} f (x) = L$ and is read as "the limit of $f (x)$ as $x$ tends to $c$ ."

The equals sign is a bit deceptive. It is not (always) a true equality, with a "limit expression" on one side, and $L$ on the other. Instead, one should think of it as a three-place relation between $f$ , $c$ , and $L$ .

History

The concept of a function's limit has been around for thousands of years, but the concept wasn't formalised until the late 19th century, when Leibniz and Newton each independently developed the limit and calculus.[cite:@openstax2020calculus]

Definition

Limits can be defined in a variety of ways, but the (ε,δ)-definition is the most well known.

Evaluation

Coming from an algebra or trigonometry class, evaluating limits requires no fundamentally-new approach to mathematics. A few elementary solutions are understood, and they are used to build more complex laws that can be composed to solve more and more limits

Limit laws

The following is a list of the most essential limit theorems:

Limits of polynomial and rational functions

The limits of polynomial functions and rational functions are very boring. The equation $\lim_{x \to c} f x = f c$ holds for all polynomial functions, and for all rational functions (given that $f c$ is defined).[cite:@openstax2020calculus]

Extensionality

Limits support a powerful version of extensionality. Rather than requiring the inner expressions of two limits to be exactly equal, limits only require equality over some open interval spanning the limit point, excluding the limit point.[cite:@openstax2020calculus] This can be used to solve limits of the indeterminate form $\frac{0}{0}$ .[cite:@openstax2020calculus]

Assume some open interval $R$ containing $c$ . If for every $x \in R ∖ {c}$ , $f x = g x$ , then

$\lim_{x \to c} f x = \lim_{x \to c} g x .$

Extensionality often comes into play for indeterminate limits. Some strategies are:

If $f x$ and $g x$ are polynomials, factoring each function and eliminating common factors. This yields a function that is equal to the original function except it will be defined at the limit point.
If the $f x$ or $g x$ contain a difference involving a square root, multiplying by the conjugate of the expression involving the square root. Usually, you will want to simplify the product involving the original square root, but not the other expression. This is to make the factors obvious.Example: in $\lim_{x \to 25} \frac{\sqrt{x} - 5}{25 - x}$ , you want to multiply by the conjugate $\frac{\sqrt{x} + 5}{\sqrt{x} + 5}$ and simplify only the numerator, leaving you with $\lim_{x \to 25} \frac{x - 25}{(25 - x) (\sqrt{x} + 5)}$ which can be easily solved.

Existence

Not every limit exists. If a function does not approach a single defined value at the point of the limit, the limit's value is undefined. For example, the function $x^{- 1}$ may appear to approach $- \infty$ at $x = 0$ , but this is only the case when looking at the left-hand side of the graph; on the right-hand side, it appears to approach $\infty$ .

Conventionally the value of a nonexistent limit is written as $DNE$ , for "does not exist."

Cases where a limit exists on only one side are still useful, albeit not proper limits. The weaker statements are called one-sided limits.