What is LHospitals Rule

LHospitals Rule is a method for finding the value of certain kinds of limits using derivatives. The rule is named after Guillaume de lHospital (or lHpital), which is a French name, pronounced low-pee-tal (NOT le Hoss-pih-tal). Guillaume de lHospital LHospitals Rule If f(x)/g(x) has the form 0/0 or / when x = a is plugged in, then: In other words, take the derivative of the numerator (top) and the derivative of the denominator (bottom), and then try computing the limit. Using LHospitals Rule In order to use lHospitals Rule, you must first check to see that your limit has the right form. First of all, it must be a fraction of two functions, f(x) / g(x) in order to apply the rule. Secondly and this is crucial! when you plug in the given x-value, the fraction must either evaluate to 0/0 or /. These are two types of indeterminate forms. If your limit problem is not in an indeterminate form, then you cant use this method directly. Examples Lets see how lHospitals Rule works in the following two examples. Example 1 After plugging in x = 0, we find the indeterminate form, 0/0. So lHospitals Rule can be used. Just take the derivative of the top and the derivative of the bottom. Afterwards, try plugging in the x value again. Example 2 LHospitals Rule works just as well in limits as x . Notice, the indeterminate form this time is /. But theres another interesting feature about this example. After using the rule once, the limit still has indeterminate form (/). Therefore, we can use the rule once again. In general, lHospitals Rule may be repeated as many times as necessary, as long as there is an indeterminate form at each stage. Functions that are not Fractions Sometimes a limit problem comes along that seems impossible to do. Standard algebraic techniques may not work. If the function had a fractional form, then we could use lHospital. But what if the function is not even a fraction? There are certain algebraic manipulations that can force an expression to be a fraction. When done correctly, lHospital may be used on the result. Forcing a Fraction Here is an example in which we change a product into a fraction using a standard algebraic trick. Example 3 This time we didnt start with a fraction. But if you rewrite x2 = 1/x -2 using negative exponents, then we can force the function to take the form of a fraction. Then use lHospitals Rule on the result. Conclusion When a limit problem involves a fraction of two functions, then try LHospitals Rule. Dont forget to check whether its an indeterminate form first. With this powerful tool in your toolbox, limits on the AP Calculus exams just got a whole lot easier!