What is the L' hospital rule in limits and derivative
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So, L'Hospital's Rule tells us that if we have an indeterminate form 0/0 or all we need to do is differentiate the numerator and differentiate the denominator and then take the limit. Before proceeding with examples let me address the spelling of “L'Hospital”. The more modern spelling is “L'Hôpital”.
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L'Hôpital's rule states that for functions f andg which are differentiable on an open intervalI 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 g'(x)\neq 0} for all x in I with x ≠ c, and
{\displaystyle \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)}}.}
The differentiation of the numerator and denominator often simplifies the quotient or converts it to a limit that can be evaluated directly.
{\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty ,}
{\displaystyle g'(x)\neq 0} for all x in I with x ≠ c, and
{\displaystyle \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)}}.}
The differentiation of the numerator and denominator often simplifies the quotient or converts it to a limit that can be evaluated directly.
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