∞ L / {\displaystyle \beta \sim \beta '} L'Hôpital's rule is a general method for evaluating the indeterminate forms {\displaystyle g(x)} 0 I like to spend my time reading, gardening, running, learning languages and exploring new places. ′ / {\displaystyle f(x)} Divide all the addends that have the highest exponent by x. / {\displaystyle 0} 0 is not an indeterminate form. ". 1 x / g {\displaystyle a} 0 if x becomes closer to zero):. = 1 To solve this indeterminate form, different types of functions must be considered. , but these limits can assume many different values. f {\displaystyle g} lim ) is not an indeterminate form, since a quotient giving rise to such an expression will always diverge. The expression Specifically, if These derivatives will allow one to perform algebraic simplification and eventually evaluate the limit. , so L'Hôpital's rule applies to it. | ∞ / ). and ′ ∞ , and c {\displaystyle \textstyle \lim _{x\to c}g(x)\;=\;0} x which means that x , / 0 x / Note that {\displaystyle +\infty } {\displaystyle 1} {\displaystyle 1} 0 → 3 f | {\displaystyle a=-\infty } If the functions 2 both approaching / f where − / 0. In these cases, a particular operation can be performed to solve each of the indeterminate forms. ′ ) g = {\displaystyle \alpha '} {\displaystyle y=x{\ln {2+\cos x \over 3}}} However it is not appropriate to call an expression "indeterminate form" if the expression is made outside the context of determining limits. ∼ c 3 cos ( c which arise from substituting {\displaystyle x} g f is an indeterminate form. x {\displaystyle 1/0} a 0 = ∞ go to , ⁡ , f / In many cases, algebraic elimination, L'Hôpital's rule, or other methods can be used to manipulate the expression so that the limit can be evaluated.. / β − This indeterminate form can be solved another way but the following must be taken into account: If the numerator and denominator have the same degree, the limit is the quotient of the coefficient of powers of the highest grade. α approaches into any of these expressions shows that these are examples correspond to the indeterminate form . {\displaystyle c} / ) x {\displaystyle a=+\infty } More specifically, an indeterminate form is a mathematical expression involving ( For more, see the article Zero to the power of zero. . {\displaystyle \lim _{x\to c}f(x)^{g(x)}} and β To see why, let ∞ {\displaystyle g} 0 = with f ∞ An indeterminate form is a limit that is still easy to solve. 0 {\displaystyle 0} ∞ ( 0 , one of these forms may be more useful than the other in a particular case (because of the possibility of algebraic simplification afterwards). 1 + a for , ′ 0 c = {\displaystyle \lim _{x\to c}{f(x)}=0,} ( ∞ In the first limit if we plugged in $$x = 4$$ we would get 0/0 and in the second limit if we “plugged” in infinity we would get $${\infty }/{-\infty }\;$$ (recall that as $$x$$ goes to infinity a polynomial will behave in the same fashion that its largest power behaves). / Infinity over Infinity To solve this indeterminate form, different types of functions must be considered. ) {\displaystyle \infty /\infty } and {\displaystyle x^{2}/x} 1 {\displaystyle 0/0} = {\displaystyle f(x)>0} {\displaystyle (1/g)/(1/f)} The indeterminate form ; if g approaches × In a loose manner of speaking, L in all three cases). {\displaystyle 0} y 0 To know the value of the limit we will have to look at the functional form of … is used in the 4th equality, and ′ is used in the 5th equality. {\displaystyle \ln L=\lim _{x\to c}({g(x)}\times \ln {f(x)})=\infty \times {-\infty }=-\infty ,} {\displaystyle 0/0} x g f One can change between these forms, if necessary, by transforming 0 lim / x In both of these cases there are competing interests or rules and it’s not clear which will win out. = y and 0 ) = ⁡ ∞ approaches y x a {\displaystyle f(x)^{g(x)}} , which is undefined. / β / / ( , one can make use of the following facts about equivalent infinitesimals (e.g., ⁡ ) / ⁡ x {\displaystyle 1-\cos x\sim {x^{2} \over 2}} f → | + β − ∞ y When two variables ≠ , / 1 {\displaystyle 0^{\infty }} = β {\displaystyle \infty } / , obtained by applying the algebraic limit theorem in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity (if a limit is confirmed as infinity, then it is not interminate since the limit is determined as infinity) and thus does not yet determine the limit being sought. g f {\displaystyle 1/0} x x ∞ , and | 1 , ( is asymptotically positive. ) → ) {\displaystyle 0^{0}} ( Moreover, if variables f x 0 can also be obtained (in the sense of divergence to infinity): The following limits illustrate that the expression ( = y In the 2nd equality, {\displaystyle 0/0} ∞ g / and 0 c α Similarly, any expression of the form {\displaystyle f(x)=|x|/(|x-1|-1)} ( For exponential functions, divide by the highest exponential base. α and / , then 0 So, given that two functions x {\displaystyle f} α  Otherwise, use the transformation in the table below to evaluate the limit.