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Indeterminate Forms

When you do direct plugging in to solve a limit, you often get $\frac00$ or $\frac\infty\infty$. Sometimes you can even get things as whacky as $0^0$, $0^\infty$, $\infty^0$, or $\infty-\infty$. These are called indeterminate forms because we can make these limits equal to whatever we want. For example, we can have

$$\lim_{x\to0}\frac{x}{x}=1$$

or

$$\lim_{x\to0}\frac{5x}{x}=5$$

or

$$\lim_{x\to0}\frac{\pi x}{x}=\pi$$

and we can get any result we want!

However, there are some very distinct forms that are NOT indeterminate forms. For example, if you plug in and get $\frac10$, this is not an indeterminate form because the limits can only be $\pm\infty$. Also, $\infty^\infty$ is not indeterminate because it is $\infty$.

L'Hospital's Rule can only be applied when you see an indeterminate form. Otherwise, if direct plugging in does not work, the limit does not exist. L'Hospital's Rule can be applied over and over to a limit until direct substitution does not yield an indeterminate form.

Most of the time, we can simplify mathematical expressions and give them a numerical value. For instance, $2+5$ can be simplified to $7$, and $\displaystyle \frac{2\times 6}{4}$ can be reduced to $3$.

However, some expressions cannot be easily given a numerical value. These expressions are labeled as "indeterminate". One common example of an indeterminate form is $\displaystyle \frac{0}{0}$. Usually, any number divided by itself equals $1$, but anything divided by $0$ should be undefined. Therefore, because we cannot determine the actual value of $\displaystyle \frac{0}{0}$, we say it is indeterminate.

In total, there are $7$ indeterminate forms involving $\infty$, $0$, and/or $1$. They are $$\frac{0}{0}, 0^0, 0\times \infty, \frac{\infty}{\infty}, \infty-\infty, \infty^{0}, 1^{\infty}.$$