Calculus: Early Transcendentals, Chapter 4, 4.4, Section 4.4, Problem 52

lim_(x->0)cot(x)-1/x
=lim_(x->0)cos(x)/sin(x)-1/x
=lim_(x->0)(xcos(x)-sin(x))/(xsin(x))
Apply L'Hospital rule , Test condition:0/0
=lim_(x->0)((xcos(x)-sin(x))')/((xsin(x))')
=lim_(x->0)(x(-sin(x))+cos(x)-cos(x))/(xcos(x)+sin(x))
=lim_(x->0)(-xsin(x))/(sin(x)+xcos(x))
Apply L'Hospital rule , Test condition:0/0
=lim_(x->0)(-xcos(x)-sin(x))/(cos(x)-xsin(x)+cos(x))
=lim_(x->0)(-sin(x)-xcos(x))/(2cos(x)-xsin(x))
=lim_(x->0)(sin(x)+xcos(x))/(xsin(x)-2cos(x))
=(sin(0)+0cos(0))/(0sin(0)-2cos(0))
=0

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