Indeterminate Forms: List & Examples

[PFuser1232]

Here's a list of all the indeterminate forms I'm familiar with:
$\frac{0}{0}$, $\frac{\infty}{\infty}$, $0⋅\infty$, $\infty - \infty$, $0^0$, $1^{\infty}$, $\infty^0$
Suppose we want to evaluate the limit:
$$\lim_{x→0} x^2 \cos{\frac{1}{x}}$$
We can find the value of this limit by applying the squeeze theorem. The limit would otherwise be indeterminate; if we plug in $x = 0$, we get:
$$0⋅\cos{\frac{1}{0}}$$
Under what category does this indeterminate form lie?

 

[ShayanJ]

This is not an indeterminate form. It has a unique limit: 0.
The reason is, whatever the argument of the cosine is, we know that the result lies between -1 and 1. So we know that we are multiplying zero by a number between -1 and 1 which is obviously zero.

 

[PFuser1232]

This is not an indeterminate form. It has a unique limit: 0.
The reason is, whatever the argument of the cosine is, we know that the result lies between -1 and 1. So we know that we are multiplying zero by a number between -1 and 1 which is obviously zero.

This is not an indeterminate form. It has a unique limit: 0.
The reason is, whatever the argument of the cosine is, we know that the result lies between -1 and 1. So we know that we are multiplying zero by a number between -1 and 1 which is obviously zero.

I'm confused. The limit of, say, $x^3 - x^2$ as $x$ approaches infinity is also unique; it's infinity. However, at first glance, it seems to be indeterminate ($\infty - \infty$).
I only arrived at the result by factoring out $x^3$.
Perhaps the definition of "indeterminate" is not so clear to me.
Could you please elaborate further?

 

[ShayanJ]

I'm confused. The limit of, say, $x^3 - x^2$ as $x$ approaches infinity is also unique; it's infinity. However, at first glance, it seems to be indeterminate ($\infty - \infty$).
I only arrived at the result by factoring out $x^3$.
Perhaps the definition of "indeterminate" is not so clear to me.
Could you please elaborate further?

The difference is that, in the case of indeterminate forms, at first you encounter one of the seven forms. Then you should find a way to see what's the value of the indeterminate form in this particular example. Its sometimes finite and sometimes infinite and sometimes doesn't exist. About your example, we first encounter $\infty - \infty$ which is indeterminate i.e. we don't know its value. Then we search for an alternative way to find its value and we figure out that because $x^3$ rises faster than $x^2$, at infinity they should be very far apart which means the answer is infinity.
But about $\displaystyle \lim_{x\to 0} x^2 \cos{\frac 1 x}$, we can directly find out that the answer is zero. Because cosine is always between -1 and 1, and here is no different. We just don't know what is the value of the cosine but we know its between -1 and 1. So just call its value a. Then we have $\displaystyle \lim_{x\to 0} x^2 a$ which is obviously zero.

 

[ShayanJ]

This page may be helpful.

 

[HallsofIvy]

I'm confused. The limit of, say, $x^3 - x^2$ as $x$ approaches infinity is also unique; it's infinity. However, at first glance, it seems to be indeterminate ($\infty - \infty$).
I only arrived at the result by factoring out $x^3$.
Perhaps the definition of "indeterminate" is not so clear to me.
Could you please elaborate further?

This particular function has limit, as x goes to infinity of 0, but there are other functions, such as $x^2- (x^2+ 1)$, that are "of the form" $\infty- \infty$ that converge to other limits. "$\infty- \infty$" and "0/0" are "indeterminate" because you cannot determine such a limit by just setting $x= 0$ or $x= \infty$ in the sequence.