L'Hopital's Rule for evaluating limits from "summary" of Differential Calculus by S Balachandra Rao
L'Hopital's Rule for evaluating limits is an important tool in calculus that allows us to evaluate limits of certain indeterminate forms. These indeterminate forms arise when we have limits that result in expressions like 0/0 or infinity/infinity, which are not immediately solvable using direct substitution. The rule states that if we have a limit of the form f(x)/g(x) as x approaches a certain value, and both f(x) and g(x) approach 0 or infinity as x approaches that value, then the limit of the ratio is equal to the limit of the ratio of the derivatives of f(x) and g(x) as x approaches the same value. In other words, we can differentiate the numerator and the denominator separately and then take the limit of the resulting expression. To apply L'Hopital's Rule, we first check if the limit we are trying to evaluate is in an indeterminate form. If it is, we differentiate the numerator and the denominator separately with respect to x. We then take the limit of the new expression as x approaches the given value. If the limit still results in an indeterminate form, we can repeat the process by differentiating the new numerator and denominator again until we reach a determinate form. It is important to note that L'Hopital's Rule can only be applied when both the numerator and the denominator approach 0 or infinity as x approaches the given value. Additionally, the rule is not applicable if the limit is not in an indeterminate form initially.- L'Hopital's Rule provides a powerful technique for evaluating limits of indeterminate forms in calculus. By differentiating the numerator and denominator separately, we can simplify complex limit problems and find the desired limit with ease.
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