L’Hospital Rule Proof

The L’Hospital rule is applied when limits result in indeterminate form 0/0, ±∞/±∞. We can prove the L’Hospital rule by using Cauchy’s Mean Value Theorem.

Let f(x) and g(x) be two continuous functions on the interval [a, b] and differentiable on the interval (a, b) and we know for any function f(x), its derivative at x = c is given as

f'(c) = [f(x) – f(c)]/(x – c)

Assume that g(x) = 0 on (a, b), then there exists c in (a, b) such that

lim _x→b [f'(x) / g'(x)] = lim _x→b [{(f(x)- f(b)) / (x – b)} / {(g(x)- g(b)) / (x – b)}]

Let the functions f and g be differentiable at x = c satisfying where c belongs to the interval in which functions are defined. Let f(c) = g(c) = 0.

By Cauchy’s Mean Value Theorem,

lim_x→c [f(x) / g(x)] = lim _x→c[f(x) -f(c)]/[g(x) – g(c)]

⇒ lim _x→c [f(x) / g(x)] = lim _x→c[f(x) – 0] / [g(x) – 0] {as f(c) = g(c) = 0}

⇒ lim _x→c [f(x) / g(x)] = lim _x→c[f(x) – f(c)] / [g(x) – g(c)]

⇒ lim_x→c [f(x) / g(x)] = lim _x→c [{(f(x)- f(c)) / (x – c)} / {(g(x)- g(c)) / (x – c)}]

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

Which is the required result.

Learn more about Mean Value Theorem.

L’ Hospital Rule in Calculus

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.

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