Rules about Limits and Infinity

[Jason76]

Are these right? These would come up after substitution, on the last step of the limit problem.

• \(\displaystyle \dfrac{n}{\infty} = 0\) Where \(\displaystyle n>0\)
• \(\displaystyle \dfrac{n}{-\infty} = 0\) Where \(\displaystyle n>0\)
• \(\displaystyle \infty(\infty) = \infty\)
• \(\displaystyle \infty + \infty = \infty\)
• \(\displaystyle \infty + n = \infty\) Where \(\displaystyle n>0\)
• \(\displaystyle \infty - \infty\) is indeterminate
• \(\displaystyle \dfrac{-\infty }{-\infty }\) is indeterminate
• \(\displaystyle \dfrac{n}{0} = \infty\)

 

[daon2]

Infinity is not a number in the ordinary sense. Stating for example \(\displaystyle n/\infty=0\) is an "intuitive" answer but I would mark it wrong if it were handed in. Thinking along these lines are to lead you to an answer rigorously, and not to just state an answer. In other words, the first two are better stated together as:

\(\displaystyle \displaystyle \text{If }\, n,A \in \mathbb{R}\, \text{ and } \,\lim_{x\to A} f(x) = \pm \infty \,\text{ then }\, \lim_{x\to A} \dfrac{n}{f(x)} = 0.\)

Also, \(\displaystyle n/0=\infty\) is wrong, even intuitively. The function \(\displaystyle 1/x\) will show you this, letting x approach zero from the right and left give you different signs of divergence.

 

[Jason76]

Infinity is not a number in the ordinary sense. Stating for example \(\displaystyle n/\infty=0\) is an "intuitive" answer but I would mark it wrong if it were handed in. Thinking along these lines are to lead you to an answer rigorously, and not to just state an answer. In other words, the first two are better stated together as:

\(\displaystyle \displaystyle \text{If }\, n,A \in \mathbb{R}\, \text{ and } \,\lim_{x\to A} f(x) = \pm \infty \,\text{ then }\, \lim_{x\to A} \dfrac{n}{f(x)} = 0.\)

Also, \(\displaystyle n/0=\infty\) is wrong, even intuitively. The function \(\displaystyle 1/x\) will show you this, letting x approach zero from the right and left give you different signs of divergence.

You said "where n is an element of real numbers", so the limit of \(\displaystyle 0\) over \(\displaystyle \infty\) and a negative number over \(\displaystyle \infty\) would also be 0.

 

[JeffM]

You said "where n is an element of real numbers", so the limit of \(\displaystyle 0\) over \(\displaystyle \infty\) and a negative number over \(\displaystyle \infty\) would also be 0.

\(\displaystyle \displaystyle n < 0\ and\ n \in \mathbb R \implies\lim_{x \rightarrow \infty}\dfrac{n}{x} = 0.\)

\(\displaystyle \displaystyle n = 0 \implies\lim_{x \rightarrow \infty}\dfrac{n}{x} = 0.\)

\(\displaystyle \displaystyle n > 0\ and\ n \in \mathbb R \implies\lim_{x \rightarrow \infty}\dfrac{n}{x} = 0.\)

\(\displaystyle \displaystyle n < 0\ and\ n \in \mathbb R \implies\lim_{x \rightarrow -\infty}\dfrac{n}{x} = 0.\)

\(\displaystyle \displaystyle n = 0 \implies\lim_{x \rightarrow -\infty}\dfrac{n}{x} = 0.\)

\(\displaystyle \displaystyle n > 0\ and\ n \in \mathbb R \implies\lim_{x \rightarrow -\infty}\dfrac{n}{x} = 0.\)

As the magnitude of x gets very large, the magnitude of n / x gets very small and so approaches zero.