Lim x/tan(x) as x approaches 0?

Question

Judging by the graph, the answer is 1. How do you solve it without graphing?

Answer 1

lim x/tan(x) = lim x*cos(x)/sin(x) = lim cos(x)/[sin(x)/x]

=lim cos(x)/lim[sin(x)/x]

lim [sin(x)/x] = 1 as x approaches 0

lim cos(x) = 1 as x approaches 0

so lim x/tan(x) = 1

Answer 2

x/tanx=

xcosx/sinx

lim (xcosx/sinx)=

x->0

lim [(x/sinx)(cosx)]=

x->0

1*1=

1