L HOSPITALS RULE EXPLANATION WITH EXAMPLES AND FULL SOLUTION AND INDETERMINATE FORM.

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# Mathematics # Limits # Indeterminate Form # Differentiation # Examples with Solution # Explanation with various reasons # Entrance Exam # Competitive Exam # Trigonometry.

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In mathematics, and more specifically in calculus, L'Hôpital's rule or L'Hospital's rule (French: [lopital]) uses derivatives to help evaluate limits involving indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to an expression that can be evaluated by substitution, allowing easier evaluation of the limit. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital. Although the contribution of the rule is often attributed to L'Hôpital, the theorem was first introduced to L'Hôpital in 1694 by the Swiss mathematician Johann Bernoulli.

L'Hôpital's rule states that for functions f and g which are differentiable on an open interval I except possibly at a point c contained in I, if {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty ,} {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty ,} {\displaystyle g'(x)\neq 0} {\displaystyle g'(x)\neq 0} for all x in I with x ≠ c, and {\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}} \lim _{x\to c}{\frac {f'(x)}{g'(x)}} exists, then

{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.} {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}
The differentiation of the numerator and denominator often simplifies the quotient or converts it to a limit that can be evaluated directly.