Mastering L’Hôpital’s Rule | A Step-by-Step Guide to Evaluating Limits of Ratios

Lim x approaches a (f(x)/g(x)=lim x approaches a (f'(x)/g'(x)

This is known as L’Hôpital’s rule, which provides a method to evaluate the limit of a ratio of two functions when both the numerator and denominator approach zero or infinity

This is known as L’Hôpital’s rule, which provides a method to evaluate the limit of a ratio of two functions when both the numerator and denominator approach zero or infinity. The rule states that if lim x approaches a (f(x)/g(x)) exists and is of the form 0/0 or ∞/∞, then:

lim x approaches a (f(x)/g(x)) = lim x approaches a (f'(x)/g'(x))

In other words, to find the limit of a ratio when both the numerator and denominator go to zero or infinity, we can differentiate both the numerator and denominator and then take the limit of the resulting ratio of derivatives.

Here’s how the rule works:

1. Start with the limit lim x approaches a (f(x)/g(x)).
2. If both f(x) and g(x) approach zero or infinity as x approaches a, proceed to the next step. Otherwise, L’Hôpital’s rule does not apply.
3. Differentiate both the numerator f(x) and denominator g(x) separately to obtain f'(x) and g'(x).
4. Evaluate the new ratio lim x approaches a (f'(x)/g'(x)).
5. If this new limit exists, then it is equal to the original limit lim x approaches a (f(x)/g(x)).

It is important to note that L’Hôpital’s rule is applicable only in certain cases and does not guarantee a solution for every limit problem. Additionally, if the new limit also results in an indeterminate form (e.g., 0/0 or ∞/∞), the rule can be applied iteratively until a determinate form is obtained or until it is clear that the limit does not exist. Finally, L’Hôpital’s rule can also be extended to other indeterminate forms such as ∞ – ∞ or 0 * ∞, by manipulating the expression algebraically before applying the rule.

More Answers: