L' Hospital

A central rule for the determination of the limit of series is the rule of L'Hospital. After several forming steps it might be that the rest of your sequels or functions can be represented as a fraction, e.g. [math]\displaystyle{ \lim_{x\rightarrow 0}\frac{sin(x)}{x} }[/math]. It is known that devisions with zero are forbidden or not definied.

The rule of L'Hospital allows us to determine the limit value of a term [math]\displaystyle{ \lim_{x\rightarrow x_0} \dfrac{ f(x)}{g(x)} }[/math], where [math]\displaystyle{ f(x) }[/math] and [math]\displaystyle{ g(x) }[/math] are sequels(or functions), either both have the limit zero or infintity.

Normally, it is not solvable what the limit of the fraction is, because in math limits of [math]\displaystyle{ \frac{0}{0} }[/math] and [math]\displaystyle{ \frac{\infty}{\infty} }[/math] are not defined. The rule of L'Hospital says, that you can replace the functions with their derivative. So basicly in formula: [math]\displaystyle{ \lim_{x\rightarrow x_0} \dfrac{ f(x)}{g(x)} \overset{l'H}{=} \lim_{x\rightarrow x_0} \dfrac{ f'(x)}{g'(x)} }[/math], if and only if [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] are derivable and have the same limit if [math]\displaystyle{ x }[/math] goes to [math]\displaystyle{ x_0 }[/math].

Remark 1: Don't try to make a derivative of the whole fraction!

Remark 2: If it is neccessary and the functions are continous derivable enough you can reapply L'Hospital until the fraction has a known limit (you have to check on every step if the single limits are equal!).