CH 4: Applications of Differentiation 4.4 Indeterminate Forms and L’Hospital Rule f(x) 0 In determining lim , we may run into indeterminate cases like . Similarly, in determining x→a g(x) 0 lim f(x)g(x) one may run into ∞ · 0, or other cases. If f 0(x) and g0(x) exist, then L’Hospital rule x→a can be used as follows: 0 ∞ 1. indeterminate form of type and ± : take derivative of the numerator of the fraction, and 0 ∞ divide it by the derivative of the denominator of the fraction. 2. BE CAREFUL: you should only use L’Hospital Rule, if indeed you are in the cases mentioned above, since otherwise it will give you incorrect answer f g 3. indeterminate form of type 0 · ∞: Re-write f · g as or to put you in one of the above 1/g 1/f cases 4. indeterminate form of type ∞ − ∞: try to convert the difference of the two functions into a quotient in order to use the above rule. In doing so, you may try a common denominator, rationalizing irrational expressions, writting out trig functions in terms of other trig functions 5. indeterminate form of type 00, ∞0, 1∞: use similar technique as we did for the derivative of xx, namely take the logarithm of both sides, and then take the limit. Don’t forget to then compute elim (expression) Homework: 6, 9, 11, 22, 33, 43, 49, 52, 55, 57, 59 4.5 Summary of Curve sketching In sketching the graph of a function, one should consider the following: 1. domain of the function 2. intercepts 3. symmetry 4. odd and even functions: a function f is even if f(−x) = f(x), like f(x) = x2 − 4. A function f is odd if f(−x) = −f(x), like f(x) = x3 + 5x 5. asymptotes: Vertical, Horizontal, and Slant Asymptotes. If the graph has no horizontal asymp- tote since the lim f(x) = ±∞, then one should look for the Slant Asymptotes. To find them x→±∞ use long division (see Example 6 page 313) 6. intervals of increase and decrease using first derivatives 7. local maximum and minimum, using 1st derivative test, or 2nd derivative test 8. concavities using second derivatives 9. points of inflections using the change in sign of the second derivative Homework: 5, 9, 15 1.