If both functions
f(x), g(x)
Are defined in an open interval containing point c and have a derivative in the interval, except maybe at c. And fulfill the following conditions:
- Both functions are tending to zero or both are tending to plus / minus infinity separately, that is we are getting the indeterminate forms:
\frac{0}{0}, \frac{\pm \infty}{\pm \infty}
2. In the interval the following holds:
g'(x)\neq 0
3. And the following holds
\lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}=L
Then the following holds
\lim _ { x \rightarrow c} \frac{f(x)}{g(x)}=\lim _ { x \rightarrow a} \frac{f'(x)}{g'(x)}=L
Notes:
- The point c can be infinity.
- Notice that the we derive the numerator and denominator separately. We do NOT use the quotient rule.
How to use Lopital Rule in other indeterminate forms
- The following indeterminate form
0\cdot (\pm \infty)
Can be transformed to an indeterminate form that is good for Lopital Rule in this way
0\cdot \infty = \frac{0}{\frac{1}{\infty}}=\frac{0}{0}
Or
0\cdot \infty = \frac{\infty}{\frac{1}{0}}=\frac{\infty}{\infty}
2. When getting the following indetrminate form
\infty - \infty
It is worth checking if calculating a common denominator will give one fraction that provides the indeterminate forms of Lupital Rule.
3. When getting the following indeterminate forms
1^{\infty}, 0^0, {\infty}^0
One can use the formula
x=e^{\ln x}
In order to get
{f(x)}^{g(x)}=e^{\ln {f(x)}^{g(x)}}=
=e^{g(x)\ln f(x)}=
Then enter the limit to the power and get a limit of a multiplication
e^{\lim _ { x \rightarrow c} {g(x)\ln f(x)}}
Press here for exercises and solution using Lopital Rule
Have a question? Found a mistake? – Write a comment below!
Was it helpful? You can buy me a cup of coffee here, which will make me very happy and will help me upload more solutions!