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About LinkBacksDefine 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 limitif 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.
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