# Elementary Infinite Limit Question

Suppose $f:\mathbb{R}\to\mathbb{R}$ is a continuous odd function. Does the following statement hold (assuming the limits exist):

$$\lim_{x\to-\infty}f(x)=-\lim_{x\to\infty}f(x)$$

This certainly seems to be true to me, because we can make this statement for any arbitrarily large, finite $x$. However, I just wanted to ensure that this statement holds in the infinite case.

• Since $f(-x) = -f(x)$ and negation is continuous, then yes. Oct 29, 2015 at 5:27
• If either limit exists then so does the other and they are the same. However, the limits may not exist. The continuity part has no bearing on the question. Oct 29, 2015 at 5:27
• Suppose the limits are infinite. Oct 29, 2015 at 5:29
• Things are OK also if we allow $\pm \infty$ as limits. But for example if $f(x)=\sin x$ then neither limit exists. Oct 29, 2015 at 5:33
• Great. That agrees with my previous intuition. Thanks for the prompt responses. Oct 29, 2015 at 5:37

Let's say:

$$\lim_{x\to~+\infty}f(x)=L$$

It means that:

$$\forall \epsilon>0,~~~~~ \exists M>0, x>M \Rightarrow |f(x)-L|<\epsilon$$

Since $f(-x)=-f(x)$

$$\forall \epsilon>0,~~~~~ \exists M>0, x>M \Rightarrow |-f(-x)-L|<\epsilon$$

or

$$\forall \epsilon>0,~~~~~ \exists -M<0, x<-M \Rightarrow |f(-x)-(-L)|<\epsilon$$

$$\forall \epsilon>0,~~~~~ \exists N<0, x<N \Rightarrow |f(-x)-(-L)|<\epsilon$$

or

$$\lim_{x\to~-\infty}f(x)=-L$$

Clearly $$\lim_{x \to -\infty}f(x) = \lim_{y \to \infty}f(-y) = \lim_{y \to \infty}(-f(y)) = -\lim_{x \to \infty}f(x)$$ The above holds without any regard to continuity of $f$ and also without any regard to the existence of the limit.

However, when the limit does not exist we need to interpret the equality in a different manner. Thus we have the following options:

• If $\lim_{x \to -\infty}f(x)$ exists then $\lim_{x \to \infty}f(x)$ also exists and $\lim_{x \to -\infty}f(x) = -\lim_{x \to \infty}f(x)$.
• If $f(x) \to \infty$ as $x \to -\infty$ then $f(x) \to -\infty$ as $x \to \infty$.
• If $f(x) \to -\infty$ as $x \to -\infty$ then $f(x) \to \infty$ as $x \to \infty$.
• If $f(x)$ oscillates finitely as $x \to -\infty$ then $f(x)$ also oscillates finitely as $x \to \infty$.
• If $f(x)$ oscillates infinitely as $x \to -\infty$ then $f(x)$ also oscillates infinitely as $x \to \infty$.