What is L’ Hospital’s Rule ?

L’ Hospital’s Rule is named after 17th-century French mathematician Guillaume De L’ Hôpital and this rule is also known as Bernoulli’s rule. This rule is used to evaluate limits using derivatives. Let’s see how to apply the rule.

 

L’ Hospital’s Rule Conditions

If the limit given, {\lim\limits_{ x \to a} \frac {f(x)} {g(x)}} is in the form of \frac {0}{0} or \frac {\infty}{\infty} and functions f(x) and g(x) both are differentiable functions then only we can apply the L’ Hospital Rule. There is one more condition that is to be checked which is ‘the derivatives must not be zero at the point x=a ‘.

 

How to apply L’ Hopital’s Rule?

As the limit {\lim\limits_{ x \to a} \frac {f(x)} {g(x)}} is in the form of \frac {0}{0} or \frac {\infty}{\infty} , we differentiate the numerator and denominator separately with respect to x or with respect to the variable we are taking the limit, here in our case the variable is x .

Let’s apply the L’ Hospital Rule

{\lim\limits_{ x \to a} \frac {f(x)} {g(x)}}   \Longrightarrow {\lim\limits_{ x \to a} \frac {f'(x)} {g'(x)}}

where f'(x) means \frac {df(x)}{dx} and g'(x) means \frac {dg(x)}{dx} .

If the limit is again in the form of \frac {0}{0} or \frac {\infty}{\infty} , then we can apply L’ Hopital’s Rule again, and so on, as below:

If {\lim\limits_{ x \to a} \frac {f'(x)} {g'(x)}} is in \frac {0}{0} or \frac {\infty}{\infty} form then

{\lim\limits_{ x \to a} \frac {f'(x)} {g'(x)}} = {\lim\limits_{ x \to a} \frac {f''(x)} {g''(x)}}

 

Solve the following questions based on L’ Hopital’s Rule or L’ Hospital’s Rule or Bernoulli’s rule

Solve: IIT JEE Advanced 2025 Paper-2 Limits Question based on L’Hospital’s Rule

Solve:IIT JEE Advanced 2025, Paper-1, Marks (+4, 0); Limits Question on Leibniz Rule

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