Question #1954

Evaluate the limit. (If you need to use -infinity or infinity, enter -INFINITY or INFINITY.)
limit x->1( (5x/x-1) -(5/ln(x))

Expert's answer

limit_{x->1}( (5x/x-1) -(5/ln(x)) = limit_{x->1}((5xlnx - 5x+5)/(x-1)lnx)

= limit_{x->1}(5xlnx - 5x+5)'/((x-1)lnx)' = limit_{x->1}(5 + 5lnx - 5)'/(1-1/x +lnx)' =

= limit_{x->1}(5/x) / ((x-1)/x^{2}) = 5/0 = ∞

= limit

= limit

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