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 ± ∞ .

Illustration:

        Evaluate  limx→1 (1-x+lnx)/(1+cos π x )

Solution:

  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

Illustration:     

Evaluate limx→y  (xy-yx)/(xx-yy )

Solution:            

          limx→y  (xy-yx)/(xx-yy );   [0/0] = limx→y (yxy-1 - yx log y)/(xx log(ex) )

                                                     = (1-log y)/log(ey)

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