Limit Divergence Criteria

This page is intended to be a part of the Real Analysis section of Math Online. Similar topics can also be found in the Calculus section of the site.

Limit Divergence Criteria

Recall from The Sequential Criterion for a Limit of a Function page, that for a function $f : A \to \mathbb{R}$ and for $c$ being a cluster point of $A$, then $\lim_{x \to c} f(x) = L$ if and only if for all sequences $(a_n)$ from $A$ for which $\lim_{n \to \infty} a_n = c$ we also have that $\lim_{n \to \infty} f(a_n) = L$. We will now formulate what is known as the Limit Divergence Criteria, which will establish criteria to establish whether the value $L$ is not the limit of $f$ as $x \to c$.

 Theorem 1 (Limit Divergence Criteria): Let $f : A \to \mathbb{R}$ be a function and let $c$ be a cluster point of $A$. Then for $L \in \mathbb{R}$, $\lim_{x \to c} f(x) \neq L$ if either: 1. There exists a sequence $(a_n)$ from $A$ where $a_n \neq c$ $\forall n \in \mathbb{N}$ such that $\lim_{n \to \infty} a_n = c$ but $\lim_{n \to \infty} f(a_n) \neq L$. 2. There exists a sequence $(a_n)$ from $A$ where $a_n \neq c$ $\forall n \in \mathbb{N}$ such that $\lim_{n \to \infty} a_n = c$ but $(f(a_n))$ does not converge in $\mathbb{R}$.
• Proof: Let $f : A \to \mathbb{R}$ be a function and let $c$ be a cluster point of $A$. Let $L \in \mathbb{R}$.
• If either (1) or (2) holds, then we note that this contradicts The Sequential Criterion for a Limit of a function. Since in both cases 1 and 2, not all sequences $(a_n)$ where $a_n \neq c$ $\forall n \in \mathbb{N}$ from $A$ such that $\lim_{n \to \infty} a_n = c$ satisfy $\lim_{n \to \infty} f(a_n) = L$, and so we have that $\lim_{x \to c} f(x) \neq L$. $\blacksquare$

The following diagram represents the first scenario of the Limit Divergence Criteria for which there exists a sequence, in this case $(b_n)$ that converges to $c$, however $(f(b_n))$ does not converge to $L$ and so $\lim_{x \to c} f(x) \neq L$: The succeeding diagram represents the second scenario of the Limit Divergence Criteria for which there exists a sequence, $(b_n)$ that converges to $c$, however $(f(b_n))$ does not converge in $\mathbb{R}$ and so $\lim_{x \to c} f(x) \neq L$: Example 1

Using the Divergence Criteria, for the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \frac{1}{x - 1}$ show that $\lim_{x \to 1} \frac{1}{x - 1}$ does not exist in $\mathbb{R}$.

Consider the sequence $(a_n) = \frac{1}{n} + 1$. We note that this sequence $(a_n)$ is from $\mathbb{R}$ (the domain). Also notice that this sequence converges to $1$ as $\lim_{n \to \infty} \frac{1}{n} + 1 = 1$. Now notice that $(f(a_n)) = \frac{1}{\frac{1}{n} + 1 - 1} = n$, and $\lim_{n \to \infty} f(a_n) = \lim_{n \to \infty} n = \infty$.

Therefore, we have a sequence $(a_n)$ from the domain that converges to $1$, however the sequence $(f(a_n))$ does not converge in $\mathbb{R}$. By (2) of the Divergence Criteria, we have that $\lim_{x \to 1} \frac{1}{x - 1}$ does not exist in $\mathbb{R}$.