Define a function f by f(x)=x+2ln(x) and let g(x)=f(x)-A-B(x-1)-C(x-1)^2. Find the values of A, B, and C so that L=lim x→1 g(x)/(x-1)^3 exists and is finite.
L'Hospital's Rule states that a limit if the limit is indeterminate, that is,
Here, if we substitute x=1 into the function , we get . Notice that if the numerator is NOT zero, then the limit will run off to . Therefore, it is necessary that A=1 so the limit is indeterminate, otherwise the limit will not converge at all. Apply L'Hospital's Rule...
Now , so repeat the process to find B. Keep going and you can get A,B,C, and also the final value of the limit L.