What I mean to say is, we know that:
$$\lim_{x\to\infty} \biggl(1+\frac{1}{x}\biggr)^x = e$$
But,
$$\lim_{x\to\infty} \biggl(\frac{1}{x}\biggr) = 0$$
Thus,
$$\lim_{x\to\infty} (1+0)^x = e$$
Finally, $1^{\infty} = e$.
Why is this incorrect? Can anyone explain where the math is wrong?
On
$1^\infty$ is indeterminate form. So $1^\infty=e$ may be true, but it may also happen that $1^\infty\ne e$
For example take $\alpha>0$ a real number and write:
$$\lim_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^{\alpha x}=e^\alpha$$
So you might as well conclude that $1^\infty=e^\alpha$
On
It's not necessarily incorrect. Notice that if we take the log of $1^\infty$, we have
$$\ln1^\infty=\infty\ln1=\infty\times0$$
And we all know very well that this indeterminate form can equal anything.
However, you reasoning is still wrong, since one may not move limits around like that.
On
Fundamentally, the problem is that you're treating $\infty$ as though it's a real number when it's really not.
On
When we state that $1^\infty$ is an indeterminate form, we mean that if $f$, $g$ are two real valued functions such that $x_0$ is in the intersection of their domains and $$\lim_{x\to x_0}f(x)=1 \qquad \lim_{x\to x_0}g(x)=+\infty \tag{1}$$ then the limit $$\lim_{x\to x_0}f(x)^{g(x)}$$ is not uniquely defined by the condition $(1)$.
In the following example, we show that for every $a>0$ there exist functions $f_a, g_a$ and $x_0\in \mathbb {\tilde R}$ satisfying $(1)$ such that $$\lim_{x\to x_0}f_a(x)^{g_a(x)}=a$$
Take $a>0$ and define $$f_a(x)=\left (1+\frac {\log(a)}{x}\right )\qquad g(x)=x$$ As you can see $f_a$ and $g$ satisfy $(1)$ in $x_0=+\infty$ for every $a$, because $$\lim_{x\to x_0}f_a(x)^{g(x)}=\lim_{x\to \infty}\left(1+\frac {\log(a)}{x} \right)^x"="1^\infty$$ Although, taking the limit, $$\lim_{x\to x_0}f_a(x)^{g(x)}=\lim_{x\to \infty}\left(1+\frac {\log(a)}{x} \right)^x=e^{\log(a)}=a$$ so the limit of an indeterminate form $1^\infty$ can be any positive real number.
On
There is a flow in your logic, you separated the limit then you did it part by part.
This is correct on it's own: $\lim_{x\to\infty} (1+\frac{1}{x})^x = e$
This is correct on it's own as well $\lim_{x\to\infty} (\frac{1}{x}) = 0$
This part is nonesensical jumoing to conclusion:
Thus,
$$\lim_{x\to\infty} (1+0)^x = e$$
because the correct version is this $\lim_{x\to\infty} (1+0)^x = 1$
Finally, $1^{\infty} = e$. Incorrect conclusion
Why is this incorrect? Can anyone explain where the math is wrong?
Math is not wrong, the logic is wrong, limit of a function is not same as function of the limit $(1+\frac{1}{x})^x$ is a function but you are applying the limit to some parts first then to the other parts, limit must be applied the whole, not part by part.
It's a common mistake to do a limit inside a limit like that. Yes, $\lim_{x \to \infty}\frac{1}{x} = 0$, but that tells you nothing about $\lim_{x \to \infty}(1 + \frac{1}{x})^x$ - for example, $\lim_{x\to\infty}x\cdot\frac{1}{x} = \lim_{x\to\infty}\frac{x}{x} = \lim_{x\to\infty}1 = 1$, even though it looks like you can say $\lim_{x\to\infty}x\cdot\frac{1}{x} = \lim_{x\to\infty}x\cdot 0 = \lim_{x\to\infty}0 = 0$. The key thing is that saying something like "$\lim_{x \to \infty}\frac{1}{x} = 0$" isn't saying "$\frac{1}{x}$ eventually becomes zero", it's saying "$\frac{1}{x}$ eventually gets as close as you like to zero". So, for large enough $x$, $1 + \frac{1}{x}$ is very close to $1$. But when $x$ is enormous, taking a number very close to $1$ and raising it to the $x$th power takes that "closeness" and pulls it wide open - $1.0001^{100000000}$ is so large Google refuses to calculate it. It so happens that the "smallness" of $\frac{1}{x}$ and the "bigness" of raising something to the $x$th power balance out in just the right way so that $\lim_{x \to \infty}(1 + \frac{1}{x})^x = e$. But change anything and you upset the balance - $\lim_{x \to \infty}(1 + \frac{1}{2x})^x = \sqrt{e}$, while $\lim_{x \to \infty}(1 + \frac{1}{x})^{2x} = e^2$.
The point is, you can never just replace something inside a limit with its limit, unless you're deleting the limit. That's an extremely dangerous operation which will only yield the right answer if you got obscenely lucky - because when you do that, you erase all of the information about how fast the expression approaches its limit, so you sacrifice the opportunity to balance things out.