Indeterminate forms and the L'Hôpital rule
In mathematics there are arithmetic expressions that are incoherent and that do not produce any logical result, these forms of expression are considered as indeterminate forms, an example of indeterminate forms are:
Why are they called indeterminate forms?
They are indeterminate forms because they do not guarantee that the limit of a rational function exists, besides this it is also an indeterminate form because it also does not indicate what the limit represents and if the limit really exists. An example of this is:
If we solve the limit by substituting x= -1, we are left with:
We are left with a limit of the indeterminate form of the form 0/0.
Logically for this limit, we can apply any algebraic artifice and solve the indeterminacy, for example we can factor the numerator:
Therefore the factorization would be:
We substitute the factorization in the limit and we are left with:
If we simplify factors we are left with:
The result of the limit that initially gave us indeterminate, turns out that when applying an algebraic artifice of factorization it gives us -2.
However, algebraic artifices cannot always be applied, especially in functions where algebraic functions are mixed with transcendental functions, as for example:
If we substitute the limit when x tends to zero we are left with an indeterminate form of the form 0/0.
In this rational function even if we apply some algebraic factorization technique we will always have indeterminate forms, so it is very useful to apply L'Hôpital's rule.
What is L'Hôpital's rule for solving indeterminate forms of the form 0/0; ∞/∞?
L'Hôpital's rule owes its name to the French mathematician Guillaume François Antoine, Marquis de L'Hôpital who wrote the first book on differential calculus in the year 1696 where L'Hôpital's rule appears, which includes the lectures of his professor, Johann Bernoulli, where Bernoulli discusses the indeterminacy 0/0. This is the method to solve these indeterminations through successive derivatives that bears the name of L'Hôpital's rule.
L'Hôpital's rule is summarized to L'Hôpital's rule theorem 8.4 found in the book Calculus with Analytic Geometry by Larson and Hostetler on page 568 which reads as follows:
If f and g are derivable on an open interval (a,b) and continuous on the closed interval [a,b] such that: 𝑔′(𝑥) ≠ 0 for any x in open interval (a,b), then there exists a point c in (a,b) such that:
If such functions are derivable on the open interval (a,b) containing c, except possibly on c itself. We assume that 𝑔′(𝑥) ≠ 0 for all x on the open interval (a,b), except possibly on c itself. If the limit of f(x)/g(x) when x tends to c yields the indeterminate form 0/0, then:
That is only true if and only if the limit on the right exists or is infinite: This result also applies if the limit of f(x)/g(x) as x tends to c yields any of the indeterminate form 0/0; ∞/∞.
Example of the L'Hôpital rule
Given the following limit, solve by L'Hôpital's rule:
To apply L'Hôpital's rule we derive the numerator and then the denominator.
Derivative of the numerator:
Derivative of the denominator:
We apply the limit to the derivatives as follows:
Bibliographic Reference
Calculus with Analytic Geometry by Ron Larson, Robert, P. Hostetler and Bruce H. Edwards. Volume I. Eighth Edition. McGraw Hill. Año 2006