Why are these expressions indeterminate expressions?

Question

Why are these $1^\infty,$ $0\cdot\infty$ and $\infty^0$ indeterminate forms. Why we can't solve these expressions?

Answer 1

Do you buy that $\frac{\infty}{\infty}$ and $\frac{0}{0}$ are indeterminate? If so, it's easy to see why those are indeterminate:

\begin{align} 0\cdot\infty =& \frac{1}{\infty} \cdot \infty = \frac{\infty}{\infty} \\ 0\cdot\infty =& 0 \cdot \frac{1}{0} = \frac{0}{0} \\ 1^\infty =& e^{\ln(1)\cdot\infty} = e^{0\cdot\infty}\text{, the exponent is indeterminate} \\ \infty^0 =& e^{\ln(\infty)\cdot0} = e^{\infty\cdot0}\text{, again, the exponent is indeterminate} \end{align}

Here is a way to informally see why they would be indeterminate. You have to remember that these are really limits. So we don't actually plug in $0$ and $\infty$.

1. 1. $\lim\limits_{x\rightarrow \infty}0\cdot x = 0$
   2. $\lim\limits_{x\rightarrow 0}x\cdot \infty = ?$ What is it approaching? At every point $x > 0$ this is infinite and at every point $x < 0$ it's $-\infty$. So which is it? $0$ or $\pm\infty$? It's indeterminate and it could be none of those.
2. 1. $\lim\limits_{x\rightarrow \infty} 1^x = 1$
   2. $\lim\limits_{x\rightarrow 1}x^\infty = ?$. Again, when $x > 1$ this is infinite and when $x < 1$ (but very close to $1$, so still positive) this is $0$. Again, so which is it? $0$, $1$, or $\infty$? It's indeterminate and, again, could be none of those.
3. 1. $\lim\limits_{x\rightarrow \infty} x^0 = 1$
   2. $\lim\limits_{x\rightarrow 0}\infty^x = ?$. Now when $x > 0$ this is infinite, when $x < 0$ this is $0$. So, for the last time, which is it, $1$, $0$, or $\infty$--it's indeterminate.

Answer 2

You can't have cohesion if you accept a certain value for $0\cdot\infty$.

For example consider the expression $\lim\limits_{x\rightarrow \infty}\frac{k}{x}\cdot x= k$.

As you can see it depends on the value k. Similar arguments can be made for the other indeterminate forms you pose.