L’ Hospital’s Rule
We have dealt with problems which had indeterminate from either 0/0 or ∞/∞ .
The other indeterminate forms are ∞-∞,0,∞,00,∞0,1∞
We state below a rule, called L' Hospital's Rule, meant for problems on limit of the form 0/0 or ∞/∞ .
Let f(x) and g(x) be functions differentiable in the neighbourhood of the point a, except may be at the point a itself. If limx→a f(x) = 0 = limx→a g(x) or limx→af(x)= ∞ = ∞ g(x), then limx→a f(x)/g(x) = limx→a f' (x)/g(x) = limx→a f' (x)/g'(x) provided that the limit on the right either exists as a finite number or is ± ∞ .
Evaluate limx→1 (1-x+lnx)/(1+cos π x )
limx→1 (1-x+lnx)/(1+cos π x ) (of the form 0/0)
= limx→1 (1-1/x)/(-π sin π x) (still of the form 0/0)
= limx→1 (x-1)/(πx sin π x) (algebraic simplification)
= limx→1 1/(πx sin π x + π2 x cos π x ) (L' Hospital's rule again)
= - 1/π2
Evaluate limx→y (xy-yx)/(xx-yy )
limx→y (xy-yx)/(xx-yy ); [0/0] = limx→y (yxy-1 - yx log y)/(xx log(ex) )
= (1-log y)/log(ey)