L’Hospital Rule in Calculus

What is L’ Hospital Rule?

L’Hospital rule states that when the limit is applied to ratio of two differentiable functions whose value after putting the limit results in indeterminate form is equal to the limit of the ratio of each function’s derivatives.

What is the Formula for L’Hospital Rule?

If  lim_x→a [f(x) / g(x)] = Indeterminate form, L’Hospital Rule Formula is given by:

lim_x→a [f(x) / g(x)] = lim_x→a [f'(x) / g'(x)]

Where,

• a is any real number or infinity,
• f'(x) is derivative of f(x), and
• g'(x) is derivative of g(x) and g(x) and g(a) ≠ 0.

When we apply L’Hospital Rule?

When for the given limits after applying the limit, we get indeterminate form (mostly 0/0, ±∞/±∞) and both functions are differentiable then, we apply L’Hospital rule.

Can L’Hospital Rule be applied multiple times?

Yes, L’Hospital rule can be applied multiple times if the resultant fraction gives indeterminate form again and again.

What are Indeterminate Forms?

Indeterminate forms are the forms with the ratio of two functions whose value in indeterminate after applying the limits. Some of the indeterminate forms are: 0/0, ±∞/±∞, 0×∞, ∞-∞, 0⁰, 1^∞ .

What are the Limitations of L’Hospital’s Rule?

One of the major limitations of L’Hospital’s Rule is that it can only be applied for indeterminate forms of either 0/0 or (+/-infinity)/(+/-infinity). All the other forms first need to be converted into these forms if possible.

L’ Hospital Rule in Calculus: L’Hospital Rule is one of the most frequently used tools in entire calculus, which helps us calculate the limit of those functions that seem indeterminate forms. For many years, these indeterminate forms have been considered impossible to solve for functions, but some scholars have found out that some functions have limits which can be seen in the graph but the calculation seems to result in an indeterminate form. Hence, the L’Hospital rule is born.

In this article, we will learn about the concept of the L’Hospital Rule in detail. Other than that, this article also covers indeterminate forms, the L’Hospital Rule formula, and proofs of the L’Hospital Rule formula with examples as well.