Math 151 Spring 2010 c Benjamin Aurispa 4 8 Indeterminate Forms and L Hospital s Rule We have seen limits in the past that take the form 00 and When we encountered these we had to do something else algebra simplification factoring to be able to find the limit These types of limits are examples of indeterminate forms If lim x a 0 f x or then we can use L Hospital s Rule to find the limit g x 0 L Hospital s Rule Suppose f and g are differentiable functions If lim x a f x 0 or then g x 0 f x f 0 x lim 0 x a g x x a g x lim Notes The limit could also be of the form or f x x a g x 0 this is NOT indeterminate The limit is 0 f x x a g x 0 this is NOT indeterminate The limit will be or If lim If lim ex 1 x 0 sin 3x 1 5 4 8 lim ln x 3 x x2 2 14 4 8 lim 1 Math 151 Spring 2010 c Benjamin Aurispa Indeterminate Products If lim f x g x 0 this limit is indeterminate Why x a 1 2 x x x 7 x2 x x2 3 x x x2 lim lim lim To find the limit the goal is to write the indeterminate product in the form Rule 1 lim csc x ln 1 sin 7x x 0 2 40 4 8 lim xex x 2 0 0 or and use L Hospital s Math 151 Spring 2010 c Benjamin Aurispa Indeterminate Difference If lim f x g x this limit is indeterminate To find the limit the x a goal is once again to convert this difference into a quotient that we can use L Hospital s Rule on if necessary lim x 0 2x 1 1 sin x x Indeterminate Powers If lim f x g x is of the form 00 0 or 1 these are indeterminate These cases x a are treated by first taking the natural logarithm which will make the limit of the form 0 Then proceed as we did with indeterminate products However we must remember to undo the natural logarithm to find our final answer Note that 0 is NOT an indeterminate form A limit of this form will be 0 1 lim x 4 1 2 x x2 3 Math 151 Spring 2010 c Benjamin Aurispa 2 lim xtan x x 0 3 62 4 8 lim ex x 1 x x Summary There are 7 basic indeterminate forms 0 0 00 0 1 0 4
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