# How are these indeterminate ?

• Jul 18th 2012, 12:03 PM
pratique21
How are these indeterminate ?
0 * infinity
1^ infinity
• Jul 18th 2012, 12:27 PM
emakarov
Re: How are these indeterminate ?
Quote:

Originally Posted by pratique21
0 * infinity
1^ infinity

I guess these are indeterminate in the sense that if $\lim_{x\to\infty} f(x)=0$ and $\lim_{x\to\infty} g(x)=\infty$, then $\lim_{x\to\infty}f(x)g(x)$ can be any number. For example, for $a>0$, $\lim_{x\to\infty}\frac{1}{x}\cdot ax=a$.

Similarly, $\lim_{n\to\infty}\left(1+\frac{a}{n}\right)=1$ and $\lim_{n\to\infty}n=\infty$, but $\lim_{n\to\infty}\left(1+\frac{a}{n}\right)^n=e^a$.
• Jul 18th 2012, 12:51 PM
Soroban
Re: How are these indeterminate ?
Hello, pratique21!

Quote:

How are these indeterminate? . $0\cdot\infty,\;\;1^{\infty}$

I'll explain the first one . . .

Consider the function: . $f(x) \:=\:\sin x\cdot\frac{1}{x}\,\text{ as }x\to0.$
We see that: . $\lim_{x\to0}\left(\sin x \cdot\frac{1}{x}\right) \:=\:0\cdot\infty$

But we have a theorem that states: . $\lim_{x\to0}\frac{\sin x}{x} \:=\:1$

. . Hence, we could say: . $0\cdot\infty \:=\:1$

Now consider the function: . $g(x) \:=\:\sin2x\cdot\frac{1}{x}\,\text{ as }x\to0.$

We see that: . $\lim_{x\to0}\left(\sin2x\cdot\frac{1}{x}\right) \:=\:0\cdot\infty$

But: . $\lim_{x\to0}\frac{\sin2x}{x} \;=\;\lim_{x\to0} \left(2\!\cdot\!\frac{\sin2x}{2x} \right) \;=\; 2\cdot\lim_{x\to0}\frac{\sin2x}{2x} \;=\; 2\cdot1 \;=\;2$

. . Hence, we could say: . $0\cdot\infty \:=\:2$

In general, we can show that $0\cdot\infty$ can equal any number.

Therefore, $0\cdot\infty$ is indeterminate.

• Jul 18th 2012, 10:02 PM
pratique21
Re: How are these indeterminate ?
Thank you Emakarov and soroban. I can understand it now. So that is the also why Infinity - infinity is indeterminate as well- It can take any value.
• Jul 18th 2012, 10:34 PM
richard1234
Re: How are these indeterminate ?
@pratique, $\infty - \infty$ is also indeterminate.