{"id":6529,"date":"2021-09-27T14:38:34","date_gmt":"2021-09-27T09:08:34","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5691"},"modified":"2025-11-09T22:30:46","modified_gmt":"2025-11-09T17:00:46","slug":"what-is-the-lhospitalss-rule","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-the-lhospitalss-rule\/","title":{"rendered":"What is the L’Hospitals’s rule?"},"content":{"rendered":"\n

Overview<\/strong><\/h2>\n\n\n\n

The L’Hospital’s rule<\/a> rule was given by  Guillaume de L’H\u00f4pital. The technique discussed in this method is used to evaluate indeterminate <\/p>\n\n\n\n

Suppose we have two functions f(x) and g(x) and both are zero at x = a, then the fraction f(a)\/ g(a) is called the indeterminate form<\/a> 0\/0.<\/p>\n\n\n\n

In other words<\/p>\n\n\n\n

Let we have two functions f(x) and g(x) and-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Then<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Is an expression of the form 0\/0<\/p>\n\n\n\n

Now, Let we have two functions f(x) and g(x) and-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Then<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Is an expression of the form   , in that case we can say that f(x)\/g(x) is an indeterminate for of the type   at x = a.<\/p>\n\n\n\n

Some other indeterminate forms<\/a> are <\/p>\n\n\n\n

In case we get indeterminate form we apply L\u2019Hospital\u2019s rule.<\/p>\n\n\n\n

L\u2019 Hospital\u2019s rule for 0\/0 form-<\/strong><\/h2>\n\n\n\n

Working steps-<\/strong><\/p>\n\n\n\n

1. Check that the limits f(x)\/g(x) is an indeterminate form of type 0\/0.<\/p>\n\n\n\n

(Note- we can not apply L\u2019Hospital rule<\/a> if it is not in indeterminate form)<\/p>\n\n\n\n

2. Differentiate f and g separately.<\/p>\n\n\n\n

3. Find the limits of the derivatives .if the limit is finite ,  then it is equal to the limit of f(x)\/g(x).<\/p>\n\n\n\n

Example-1: Evaluate <\/strong><\/strong><\/p>\n\n\n\n

Sol. Here we notice that it is an indeterminate form<\/a> of .0\/0<\/p>\n\n\n\n

So that , we can apply L\u2019Hospital rule-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Note- Suppose we get an indeterminate form even after finding first derivative, then in that case , we use the other form of L\u2019Hospital\u2019s rule.<\/strong><\/p>\n\n\n\n

If we have f(x) and g(x) are two functions such that <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

If exist or (\u221e , -\u221e), then<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Example-3: Evaluate <\/strong><\/strong><\/p>\n\n\n\n

Sol. Let <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

then<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

And<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

But if we use L\u2019Hospital rule again, then we get- <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

L\u2019Hospital\u2019s rule for \u221e\/\u221e form<\/strong><\/h2>\n\n\n\n

Let f and g are two differentiable functions on an open interval containing x = a, except possibly at x = a and that<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

If has a finite limit, or if it is +\u221e or-\u221e , then<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Theorem- If we have f(x) and g(x) are two functions such that <\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

If <\/strong><\/strong> exist or (<\/strong>\u221e<\/strong> , –<\/strong>\u221e), then<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Example-5: Find <\/strong><\/strong> , n>0.<\/strong><\/p>\n\n\n\n

Sol. Let f(x) = log x and g(x) = x^n<\/p>\n\n\n\n

These two functions satisfied the theorem that we have discussed above-<\/p>\n\n\n\n

So that,<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n