Two methods? Evaluate the following limits in two different ways: Use the methods 4 of Chapter 2and use l'Hôpital's Rule. lim 4x ??x x?? 2x + x?1

Solution Step 1 4x x In this problem we have to find the value of lim 2x + x1in two different ways. x The first method is to simplify the given problem and then apply the limit and the second method is to use l’hopital’s rule. l'Hôpital's Rule: Suppose that we have one of the following cases, lim f(x)= or lim f(x= ± xa g(x) 0 xa g(x) ± Where a can be any real number, infinity or negative infinity. f(x) f(x) In these cases we have lim g(x)= lim gx) xa xa Step 2 4 Let us find the value of lim 4x xusing first method. x 2x + x1 So let us simplify the expression. 4x x 4x x Consider lim 2x + x1 = lim 4 1 x x 2x + x We have x , then 1 1 = 0 x 1 Thus using x 0we get, 4 = lim 4x x 10 2x + x x 4x x = lim 4 x 2x = lim(2 x) x 2x4 x = 2 x 2x 1 1 1 We have x x = 0 then x4 0 Thus using 14 0we get, x x = 2 lim ( 4 10 2 x x4 = 2 0 = 2 Thus by using first method we get, 4x x lim 4 1 = 2 x 2x + x