L’Hôspital Rule

It is known as Bernoulli’s rule, and it is a mathematical theorem that is used to evaluate the limits of indeterminate forms. In honor of the French philosopher L’Hôpital, who presented the rule for the first time in the 18th century, it was given this name. Using the rule (or using it again and again) typically leads to the transformation of an ambiguous form into an expression that can be evaluated by substituting the correct expression. Guillaume de l’Hôpital, a French mathematician who flourished in the 17th century, is credited with giving the rule its name. Even though L’Hôpital is widely credited with developing this rule, the theorem was originally transmitted to him in 1694 by the Swiss mathematician Johann Bernoulli, who is also credited with discovering it.

L’Hôspital’s Rule

A general method of assessing indeterminate forms such as 0/0 or ∞/∞, L’Hospital’s rule can be applied to any scenario and is useful in a variety of situations. For the derivatives of a function, L’Hospital’s rule is used to establish the limits of indeterminate forms for the derivatives of the function. The L-Hospital rule can be applied more than once in a single case if the circumstances warrant it. Continuing to apply this rule is permissible so long as it keeps any indeterminate form after it has been applied once is permitted. A problem that does not fall into one of the indeterminate forms cannot be solved by applying L’Hospital’s Rule to the situation.

L’Hôspital’s Rule Formula

The rule of the hospital stipulates that

ifxcf(x)=xcg(x)=0 or ± ∞, g'(x)≠0 for all x in I with x≠c, and

xcf'(x)g'(x) exists, then

xcf(x)g(x) = xcf'(x)g'(x)

L’Hôspital’s Rule proof

The L’Hospital’s rule can be demonstrated through the use of the Extended Mean Value Theorem or Cauchy’s Mean Value Theorem.

Pretend the two functions f and g are defined on the interval (c, b) in such a way that f(x)→0 and g(x)→0, as x→c+.

We do, however, have the fact that f'(c) / g'(c) tends to finite limits, which we can use to our advantage. In this case, the functions f and g are differentiated, and the functions f'(x) and g'(x) exist on the set [c, c+k], respectively. Additionally, the functions f’ and g’ are continuous on the interval [c, c+k], provided that the requirements f(c), g(c), and g'(c) are met.

Using the Cauchy Mean Value According to the theorem, there exists ck∈ (c, c+k), such that

f’(ck)/g’(ck) = [f(c+k)-f(c)]/[g(c+k)-g(c)] = f(c+k)/g(c+k)

Now, k→0+,

k0+f‘(ck)g‘(ck)=xc+f‘(x)g‘(x)  

While,

k0+f(c+k)g(c+k)=x0+f(x)g(x)  

So, we have

xc+f(x)g(x)=xc+f‘(x)g‘(x)  

L’Hôspital’s Rule Uses

Using L Hospital’s rule, we can answer the issue in either of the following forms: /0, ∞/∞, ∞ – ∞, 0 x /∞, 1/∞, ∞0, or 00. These types of shapes are referred to as indeterminate forms. L’Hospital’s rule can be used to the issue in order to eliminate the indeterminate forms.

Conclusion

L’Hospital’s rule is the only approach to simplify the evaluation of limits in all cases. While it does not directly examine boundaries, when utilized effectively, it only serves to simplify the evaluation process. A link between one limit (of f/g) and another limit (of f′/g′) is established by Macho L’Hospital’s Rule, but not to the value of f′(a)/g′. (a).