Limits

The concept of a limit in calculus addresses the behavior of a function close to a particular input.

Informally, a function $f(x)$ has a limit $L$ as $x$ approaches a value $a$ if $f(x)$ gets arbitrarily close to $L$ as $x$ gets sufficiently close to $a$. The value of the function at $a$, $f(a)$, does not matter when determining the limit.

The information above is lifted from a Wikipedia article.

Formal definitions¶

The informal definition of a limit is made precise by the ($\epsilon, \delta$)-definition, which is presented below.

Two-sided limit¶

The statement $\lim_{x \to a} f(x) = L$ is read as “the limit of $f(x)$ as $x$ approaches $a$ is $L$”. The statement is formally defined as:

The definition can be understood as a challenge-response game. An opponent challenges us with a small positive number $\epsilon$, which represents the desired closeness to the limit $L$. We must produce a small positive number $\delta$, which represents the required closeness to the point $a$. If we can always produce a $\delta$ for any given $\epsilon$, the limit exists.

One-sided limits¶

The statement $\lim_{x \to a^+} f(x) = L$ is read as “the limit of $f(x)$ as $x$ approaches $a$ from the right is $L$”. The statement is defined as:

The statement $\lim_{x \to a^-} f(x) = L$ is read as “the limit of $f(x)$ as $x$ approaches $a$ from the left is $L$”. The statement is defined as:

One-sided limits consider the behavior of the function as $x$ approaches $a$ from only one side. The same challenge-response game applies, but the closeness condition on $x$ is restricted to one side of $a$.

Relationship¶

The two-sided limit exists if and only if both one-sided limits exist and are equal.