# Other Indeterminate Forms

## Related calculator: Limit Calculator

We already talked about other indeterminate forms (indeterminate differences, indeterminate products and indeterminate powers), so we know that we can convert them into either indeterminate form of type $\frac{{0}}{{0}}$ or indeterminate form of type $\frac{\infty}{\infty}$. This allows us to use either First or Second L'Hopital's Rules.

Indeterminate Products

Product will have indeterminate form only if $\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}={0}$ and $\lim_{{{x}\to{a}}}{g{{\left({x}\right)}}}=\infty$ (or $-\infty$ ). In this case $\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}{g{{\left({x}\right)}}}$ is indeterminate form of type ${0}\cdot\infty$.

We can deal with it by writing product as a quotient:

$\lim_{{{x}\to{a}}}\frac{{f{{\left({x}\right)}}}}{{\frac{{1}}{{g{{\left({x}\right)}}}}}}$ (this will give indeterminate form of type $\frac{{0}}{{0}}$) or $\lim_{{{x}\to{a}}}\frac{{g{{\left({x}\right)}}}}{{\frac{{1}}{{f{{\left({x}\right)}}}}}}$ (this will give indeterminate form of type $\frac{{\infty}}{{\infty}}$).

Example 1. Find $\lim_{{{x}\to{{0}}^{+}}}{x}{\ln{{\left({x}\right)}}}$.

Since ${x}\to{0}$ and ${\ln{{\left({x}\right)}}}\to-\infty$ when ${x}\to{{0}}^{+}$ this is indeterminate form of type ${0}\cdot\infty$.

Representing product as quotient we obtain indeterminate form of type $\frac{{\infty}}{{\infty}}$ and therefore can apply Second L'Hospital's Rule:

$\lim_{{{x}\to{{0}}^{+}}}{x}{\ln{{\left({x}\right)}}}=\lim_{{{x}\to{{0}}^{+}}}\frac{{{\ln{{\left({x}\right)}}}}}{{\frac{{1}}{{x}}}}=\lim_{{{x}\to{{0}}^{+}}}\frac{{{\left({\ln{{\left({x}\right)}}}\right)}'}}{{{\left(\frac{{1}}{{x}}\right)}'}}=\lim_{{{x}\to{{0}}^{+}}}\frac{{\frac{{1}}{{x}}}}{{-\frac{{1}}{{{x}}^{{2}}}}}=$

$=\lim_{{{x}\to{{0}}^{+}}}-{x}={0}$.

Note, that we could represent product as indeterminate form of type $\frac{{0}}{{0}}$: $\lim_{{{x}\to{{0}}^{+}}}{x}{\ln{{\left({x}\right)}}}=\lim_{{{x}\to{{0}}^{+}}}\frac{{x}}{{\frac{{1}}{{{\ln{{\left({x}\right)}}}}}}}$. But applying L'Hopital's Rule to this limit will give a more complicated expression than the one we started with.

In general, when we rewrite an indeterminate product, we try to choose the option that leads to the simpler limit.

Indeterminate Differences

Difference will have indeterminate form only if $\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}=\infty$ and $\lim_{{{x}\to{a}}}{g{{\left({x}\right)}}}=\infty$. In this case limit $\lim_{{{x}\to{a}}}{\left({f{{\left({x}\right)}}}-{g{{\left({x}\right)}}}\right)}$ is indeterminate form of type $\infty-\infty$.

We can deal with it by rewriting it as quotient:

${f{{\left({x}\right)}}}-{g{{\left({x}\right)}}}=\frac{{1}}{{\frac{{1}}{{{f{{\left({x}\right)}}}}}}}-\frac{{1}}{{\frac{{1}}{{{g{{\left({x}\right)}}}}}}}=\frac{{\frac{{1}}{{{g{{\left({x}\right)}}}}}-\frac{{1}}{{{f{{\left({x}\right)}}}}}}}{{\frac{{1}}{{{f{{\left({x}\right)}}}}}\cdot\frac{{1}}{{{g{{\left({x}\right)}}}}}}}$. However, on practice there is a much simpler expression.

Example 2. Compute $\lim_{{{x}\to{{\left(\frac{\pi}{{2}}\right)}}^{{-}}}}{\left({\sec{{\left({x}\right)}}}-{\tan{{\left({x}\right)}}}\right)}$.

Note that ${\sec{{\left({x}\right)}}}\to\infty$ and ${\tan{{\left({x}\right)}}}\to\infty$ as ${x}\to{{\left(\frac{\pi}{{2}}\right)}}^{{-{}}}$. Therefore we have indeterminate for of type $\infty-\infty$.

Here we can use a common denominator:

$\lim_{{{x}\to{{\left(\frac{\pi}{{2}}\right)}}^{{-}}}}{\left({\sec{{\left({x}\right)}}}-{\tan{{\left({x}\right)}}}\right)}=\lim_{{{x}\to{{\left(\frac{\pi}{{2}}\right)}}^{{-}}}}{\left(\frac{{1}}{{{\cos{{\left({x}\right)}}}}}-\frac{{{\sin{{\left({x}\right)}}}}}{{{\cos{{\left({x}\right)}}}}}\right)}=\lim_{{{x}\to{{\left(\frac{\pi}{{2}}\right)}}^{{-}}}}\frac{{{1}-{\sin{{\left({x}\right)}}}}}{{{\cos{{\left({x}\right)}}}}}$.

We can use L'Hopital's rule now, because $\lim_{{{x}\to{{\left(\frac{\pi}{{2}}\right)}}^{{-}}}}{\left({1}-{\sin{{\left({x}\right)}}}\right)}={0}$ and $\lim_{{{x}\to{{\left(\frac{\pi}{{2}}\right)}}^{{-}}}}{\cos{{\left({x}\right)}}}={0}$:

$\lim_{{{x}\to{{\left(\frac{\pi}{{2}}\right)}}^{{-}}}}\frac{{{1}-{\sin{{\left({x}\right)}}}}}{{{\cos{{\left({x}\right)}}}}}=\lim_{{{x}\to{{\left(\frac{\pi}{{2}}\right)}}^{{-}}}}\frac{{{\left({1}-{\sin{{\left({x}\right)}}}\right)}'}}{{{\left({\cos{{\left({x}\right)}}}\right)}'}}=\lim_{{{x}\to{{\left(\frac{\pi}{{2}}\right)}}^{{-}}}}\frac{{-{\cos{{\left({x}\right)}}}}}{{-{\sin{{\left({x}\right)}}}}}={0}$.

Example 3. Calculate $\lim_{{{x}\to{0}}}{\left({{\cot}}^{{2}}{\left({x}\right)}-\frac{{1}}{{{x}}^{{2}}}\right)}$.

Clearly we have indeterminate form of type $\infty-\infty$ here.

Let's simplify this expression: ${{\cot}}^{{2}}{\left({x}\right)}-\frac{{1}}{{{x}}^{{2}}}=\frac{{{{\cos}}^{{2}}{\left({x}\right)}}}{{{{\sin}}^{{2}}{\left({x}\right)}}}-\frac{{1}}{{{x}}^{{2}}}=\frac{{{{x}}^{{2}}{{\cos}}^{{2}}{\left({x}\right)}-{{\sin}}^{{2}}{\left({x}\right)}}}{{{{x}}^{{2}}{{\sin}}^{{2}}{\left({x}\right)}}}=\frac{{{\left({x}{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}\right)}{\left({x}{\cos{{\left({x}\right)}}}-{\sin{{\left({x}\right)}}}\right)}}}{{{{x}}^{{2}}{{\sin}}^{{2}}{\left({x}\right)}}}=$

$=\frac{{{x}{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}}}{{{x}}}\cdot\frac{{{x}{\cos{{\left({x}\right)}}}-{\sin{{\left({x}\right)}}}}}{{{x}{{\sin}}^{{2}}{\left({x}\right)}}}$.

We can found limit of first factor very easy: $\lim_{{{x}\to{0}}}\frac{{{x}{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}}}{{x}}=\lim_{{{x}\to{0}}}{\left({\cos{{\left({x}\right)}}}+\frac{{{\sin{{\left({x}\right)}}}}}{{x}}\right)}={\cos{{\left({0}\right)}}}+{1}={2}$.

Limit of second factor is indeterminate form of type $\frac{{0}}{{0}}$ so we can apply L'Hopital's Rule:

$\lim_{{{x}\to{0}}}\frac{{{x}{\cos{{\left({x}\right)}}}-{\sin{{\left({x}\right)}}}}}{{{x}{{\sin}}^{{2}}{\left({x}\right)}}}=\lim_{{{x}\to{0}}}\frac{{{\left({x}{\cos{{\left({x}\right)}}}-{\sin{{\left({x}\right)}}}\right)}'}}{{{\left({x}{{\sin}}^{{2}}{\left({x}\right)}\right)}'}}=\lim_{{{x}\to{0}}}\frac{{{\cos{{\left({x}\right)}}}-{x}{\sin{{\left({x}\right)}}}-{\cos{{\left({x}\right)}}}}}{{{{\sin}}^{{2}}{\left({x}\right)}+{2}{x}{\sin{{\left({x}\right)}}}{\cos{{\left({x}\right)}}}}}=$

$=\lim_{{{x}\to{0}}}\frac{{-{x}}}{{{\sin{{\left({x}\right)}}}+{2}{x}{\cos{{\left({x}\right)}}}}}=\lim_{{{x}\to{0}}}\frac{{-{1}}}{{\frac{{{\sin{{\left({x}\right)}}}}}{{x}}+{2}{\cos{{\left({x}\right)}}}}}=\frac{{-{1}}}{{{1}+{2}\cdot{\cos{{\left({0}\right)}}}}}=-\frac{{1}}{{3}}$.

Therefore,

$\lim_{{{x}\to{0}}}{\left({{\cot}}^{{2}}{\left({x}\right)}-\frac{{1}}{{{x}}^{{2}}}\right)}=\lim_{{{x}\to{0}}}{\left(\frac{{{x}{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}}}{{{x}}}\cdot\frac{{{x}{\cos{{\left({x}\right)}}}-{\sin{{\left({x}\right)}}}}}{{{x}{{\sin}}^{{2}}{\left({x}\right)}}}\right)}=$

$=\lim_{{{x}\to{0}}}\frac{{{x}{\cos{{\left({x}\right)}}}-{\sin{{\left({x}\right)}}}}}{{{x}}}\cdot\lim_{{{x}\to{0}}}\frac{{{x}{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}}}{{{x}{{\sin}}^{{2}}{\left({x}\right)}}}={2}\cdot{\left(-\frac{{1}}{{3}}\right)}=-\frac{{2}}{{3}}$.

Indeterminate Powers

Several indeterminate forms arise from the $\lim_{{{x}\to{a}}}{{\left({f{{\left({x}\right)}}}\right)}}^{{{g{{\left({x}\right)}}}}}$.

1. $\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}={0}$ and $\lim_{{{x}\to{a}}}{g{{\left({x}\right)}}}={0}$ give indeterminate form of type ${{0}}^{{0}}$ (it is indeterminate because ${{0}}^{{x}}={0}$ for any ${x}>{0}$ but ${{x}}^{{0}}={1}$ for any ${x}\ne{0}$ ).
2. $\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}=\infty$ and $\lim_{{{x}\to{a}}}{g{{\left({x}\right)}}}={0}$ give indeterminate form of type ${\infty}^{{0}}$.
3. $\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}={1}$ and $\lim_{{{x}\to{a}}}{g{{\left({x}\right)}}}=\pm\infty$ give indeterminate form of type ${{1}}^{{\infty}}$.

Each of these three cases can be treated either by:

1. Taking the natural logarithm: let ${y}={{\left({f{{\left({x}\right)}}}\right)}}^{{{g{{\left({x}\right)}}}}}$ then ${\ln{{\left({y}\right)}}}={\ln{{\left({{\left({f{{\left({x}\right)}}}\right)}}^{{{g{{\left({x}\right)}}}}}\right)}}}={g{{\left({x}\right)}}}{\ln{{\left({f{{\left({x}\right)}}}\right)}}}$.
2. Writing the function as an exponential: ${{\left({f{{\left({x}\right)}}}\right)}}^{{{g{{\left({x}\right)}}}}}={{e}}^{{{g{{\left({x}\right)}}}{\ln{{\left({f{{\left({x}\right)}}}\right)}}}}}$.

In either method we are led to the indeterminate product ${g{{\left({x}\right)}}}{\ln{{\left({f{{\left({x}\right)}}}\right)}}}$, which is of type ${0}\cdot\infty$.

Example 4. Find $\lim_{{{x}\to{{0}}^{+}}}{{\left({1}+{\sin{{\left({2}{x}\right)}}}\right)}}^{{{\cot{{\left({x}\right)}}}}}$.

First, make sure that this is indeterminate limit: since ${1}+{\sin{{\left({2}{x}\right)}}}\to{1}$ and ${\cot{{\left({x}\right)}}}\to\infty$ as ${x}\to{{0}}^{+}$ then this is indeterminate form of type ${{1}}^{{\infty}}$.

Let ${y}={{\left({1}+{\sin{{\left({2}{x}\right)}}}\right)}}^{{{\cot{{\left({x}\right)}}}}}$ then ${\ln{{\left({y}\right)}}}={\cot{{\left({x}\right)}}}{\ln{{\left({1}+{\sin{{\left({2}{x}\right)}}}\right)}}}=\frac{{{\ln{{\left({1}+{\sin{{\left({2}{x}\right)}}}\right)}}}}}{{{\tan{{\left({x}\right)}}}}}$.

Since ${\ln{{\left({1}+{\sin{{\left({2}{x}\right)}}}\right)}}}\to{0}$ and ${\tan{{\left({x}\right)}}}\to{0}$ as ${x}\to{{0}}^{+}$, we have indeterminate form of type $\frac{{0}}{{0}}$ and can apply L'Hopital's Rule:

$\lim_{{{x}\to{{0}}^{+}}}{\ln{{\left({y}\right)}}}=\lim_{{{x}\to{{0}}^{+}}}\frac{{{\ln{{\left({1}+{\sin{{\left({2}{x}\right)}}}\right)}}}}}{{{\tan{{\left({x}\right)}}}}}=\lim_{{{x}\to{{0}}^{+}}}\frac{{{\left({\ln{{\left({1}+{\sin{{\left({2}{x}\right)}}}\right)}}}\right)}'}}{{{\left({\tan{{\left({x}\right)}}}\right)}'}}=$

$=\lim_{{{x}\to{{0}}^{+}}}\frac{{\frac{{{2}{\cos{{\left({2}{x}\right)}}}}}{{{1}+{\sin{{\left({2}{x}\right)}}}}}}}{{{{\sec}}^{{2}}{\left({x}\right)}}}={2}$.

We found limit of ${\ln{{\left({y}\right)}}}$, but we need limit of ${y}$, so

$\lim_{{{x}\to{{0}}^{+}}}{{\left({1}+{\sin{{\left({2}{x}\right)}}}\right)}}^{{{\cot{{\left({x}\right)}}}}}=\lim_{{{x}\to{{0}}^{+}}}{y}=\lim_{{{x}\to{{0}}^{+}}}{{e}}^{{{\ln{{\left({y}\right)}}}}}={{e}}^{{\lim_{{{x}\to{{0}}^{+}}}{\ln{{\left({y}\right)}}}}}={{e}}^{{2}}$.

Example 5. Calculate $\lim_{{{x}\to{{0}}^{+}}}{{x}}^{{x}}$.

Notice that this is indeterminate form of type ${{0}}^{{0}}$. We can proceed as in Example 4, but let's do it in another way:

$\lim_{{{x}\to{{0}}^{+}}}{{x}}^{{x}}=\lim_{{{x}\to{{0}}^{+}}}{{e}}^{{{\ln{{\left({{x}}^{{x}}\right)}}}}}=\lim_{{{x}\to{{0}}^{+}}}{{e}}^{{{x}{\ln{{\left({x}\right)}}}}}={{e}}^{{\lim_{{{x}\to{{0}}^{+}}}{x}{\ln{{\left({x}\right)}}}}}={{e}}^{{0}}={1}$ because $\lim_{{{x}\to{{0}}^{+}}}{x}{\ln{{\left({x}\right)}}}={0}$.

Example 6. Find $\lim_{{{x}\to{0}}}{{\left(\frac{{\sin{{\left({x}\right)}}}}{{x}}\right)}}^{{\frac{{1}}{{{1}-{\cos{{\left({x}\right)}}}}}}}$.

Here we have indeterminate form of type ${{1}}^{\infty}$.

Let ${y}={{\left(\frac{{\sin{{\left({x}\right)}}}}{{x}}\right)}}^{{\frac{{1}}{{{1}-{\cos{{\left({x}\right)}}}}}}}$. We can consider the case ${x}>{0}$ (because ${y}$ is odd function).

So, we have that ${\ln{{\left({y}\right)}}}={\ln{{\left({{\left(\frac{{\sin{{\left({x}\right)}}}}{{x}}\right)}}^{{\frac{{1}}{{{1}-{\cos{{\left({x}\right)}}}}}}}\right)}}}=\frac{{1}}{{{1}-{\cos{{\left({x}\right)}}}}}{\ln{{\left(\frac{{\sin{{\left({x}\right)}}}}{{x}}\right)}}}=\frac{{{\ln{{\left({\sin{{\left({x}\right)}}}\right)}}}-{\ln{{\left({x}\right)}}}}}{{{1}-{\cos{{\left({x}\right)}}}}}$.

Now, applying of L'Hopital's Rule gives the following:

$\lim_{{{x}\to{0}}}{\ln{{\left({y}\right)}}}=\lim_{{{x}\to{0}}}\frac{{{\ln{{\left({\sin{{\left({x}\right)}}}\right)}}}-{\ln{{\left({x}\right)}}}}}{{{1}-{\cos{{\left({x}\right)}}}}}=\lim_{{{x}\to{0}}}\frac{{{\left({\ln{{\left({\sin{{\left({x}\right)}}}\right)}}}-{\ln{{\left({x}\right)}}}\right)}'}}{{{\left({1}-{\cos{{\left({x}\right)}}}\right)}'}}=\lim_{{{x}\to{0}}}\frac{{\frac{{{\cos{{\left({x}\right)}}}}}{{{\sin{{\left({x}\right)}}}}}-\frac{{1}}{{x}}}}{{{\sin{{\left({x}\right)}}}}}=$

$=\lim_{{{x}\to{0}}}\frac{{{x}{\cos{{\left({x}\right)}}}-{\sin{{\left({x}\right)}}}}}{{{x}{{\sin}}^{{2}}{\left({x}\right)}}}=-\frac{{1}}{{3}}$ (as was found in Example 3).

So, $\lim_{{{x}\to{0}}}{y}={{e}}^{{-\frac{{1}}{{3}}}}=\frac{{1}}{{{\sqrt[{{3}}]{{{e}}}}}}$.

Example 7. Find $\lim_{{{x}\to\infty}}{{\left(\frac{\pi}{{2}}-{\operatorname{arctan}{{\left({x}\right)}}}\right)}}^{{\frac{{1}}{{\ln{{\left({x}\right)}}}}}}$.

Since $\frac{\pi}{{2}}-{\operatorname{arctan}{{\left({x}\right)}}}\to{0}$ and $\frac{{1}}{{{\ln{{\left({x}\right)}}}}}\to{0}$ as ${x}\to\infty$, we have indeterminate form of type ${{0}}^{{0}}$.

Let ${y}={{\left(\frac{\pi}{{2}}-{\operatorname{arctan}{{\left({x}\right)}}}\right)}}^{{\frac{{1}}{{{\ln{{\left({x}\right)}}}}}}}$, then ${\ln{{\left({y}\right)}}}={\ln{{\left({{\left(\frac{\pi}{{2}}-{\operatorname{arctan}{{\left({x}\right)}}}\right)}}^{{\frac{{1}}{{{\ln{{\left({x}\right)}}}}}}}\right)}}}=\frac{{1}}{{{\ln{{\left({x}\right)}}}}}{\ln{{\left(\frac{\pi}{{2}}-{\operatorname{arctan}{{\left({x}\right)}}}\right)}}}$.

So, $\lim_{{{x}\to\infty}}{\ln{{\left({y}\right)}}}=\lim_{{{x}\to\infty}}\frac{{{\ln{{\left(\frac{\pi}{{2}}-{\operatorname{arctan}{{\left({x}\right)}}}\right)}}}}}{{{\ln{{\left({x}\right)}}}}}=\lim_{{{x}\to\infty}}\frac{{{\left({\ln{{\left(\frac{\pi}{{2}}-{\operatorname{arctan}{{\left({x}\right)}}}\right)}}}\right)}'}}{{{\left({\ln{{\left({x}\right)}}}\right)}'}}=\lim_{{{x}\to\infty}}\frac{{\frac{{1}}{{\frac{\pi}{{2}}-{\operatorname{arctan}{{\left({x}\right)}}}}}\cdot{\left(-\frac{{1}}{{{1}+{{x}}^{{2}}}}\right)}}}{{\frac{{1}}{{x}}}}=$

$=\lim_{{{x}\to\infty}}\frac{{\frac{{x}}{{{1}+{{x}}^{{2}}}}}}{{{\operatorname{arctan}{{\left({x}\right)}}}-\frac{\pi}{{2}}}}$.

On this stage we obtained indeterminate form of type $\frac{{0}}{{0}}$ so we apply L'Hopital's Rule once more:

$\lim_{{{x}\to\infty}}\frac{{\frac{{x}}{{{1}+{{x}}^{{2}}}}}}{{{\operatorname{arctan}{{\left({x}\right)}}}-\frac{\pi}{{2}}}}=\lim_{{{x}\to\infty}}\frac{{{\left(\frac{{x}}{{{1}+{{x}}^{{2}}}}\right)}'}}{{{\left({\operatorname{arctan}{{\left({x}\right)}}}-\frac{\pi}{{2}}\right)}'}}=\lim_{{{x}\to\infty}}\frac{{\frac{{{1}-{{x}}^{{2}}}}{{{{\left({1}+{{x}}^{{2}}\right)}}^{{2}}}}}}{{\frac{{1}}{{{1}+{{x}}^{{2}}}}}}=\lim_{{{x}\to\infty}}\frac{{{1}-{{x}}^{{2}}}}{{{1}+{{x}}^{{2}}}}=\lim_{{{x}\to\infty}}\frac{{{{x}}^{{2}}{\left(-{1}+\frac{{1}}{{{x}}^{{2}}}\right)}}}{{{{x}}^{{2}}{\left({1}-\frac{{1}}{{{x}}^{{2}}}\right)}}}=$

$=\lim_{{{x}\to\infty}}\frac{{-{1}+\frac{{1}}{{{x}}^{{2}}}}}{{{1}+\frac{{1}}{{{x}}^{{2}}}}}=-{1}$.

So, $\lim_{{{x}\to\infty}}{{\left(\frac{\pi}{{2}}-{\operatorname{arctan}{{\left({x}\right)}}}\right)}}^{{\frac{{1}}{{{\ln{{\left({x}\right)}}}}}}}=\lim_{{{x}\to\infty}}{y}=\lim_{{{x}\to\infty}}{{e}}^{{{\ln{{\left({y}\right)}}}}}={{e}}^{{\lim_{{{x}\to\infty}}{\ln{{\left({y}\right)}}}}}={{e}}^{{-{1}}}=\frac{{1}}{{e}}$.