For a positive central angle of #x# radians (#0 < x < pi/2#) (not degrees)
Source:
The geometric idea is that
#"Area of "Delta KOA < "Area of " "Sector KOA" < "Area of "Delta LOA#
#"Area of "Delta KOA = 1/2(1)(sinx) \ \ \ # (#1/2"base"*"height"#)
#"Area of " "Sector KOA" = 1/2 (1)^2 x \ \ \ # (#x# is in radians)
#"Area of "Delta LOA = 1/2tanx \ \ \ # (#AL = tanx#)
So we have:
#sinx/2 < x/2 < tanx/2#
For small positive #x#, we have #inx > 0# so we can multiply through by #2/sinx#, to get
#1 < x/sinx < 1/cosx#
So
#cosx < sinx/x < 1# for #0 < x < pi/2#.
#lim_(xrarr0^+) cosx = 1# and #lim_(xrarr0^+) 1= 1#
so #lim_(xrarr0^+) sinx/x = 1#
We also have, for these small #x#, #sin(-x) = -sinx#, so #(-x)/sin(-x) = x/sinx# and #cos(-x) = cosx#, so
#cosx < sinx/x < 1# for #-pi/2 < x < 0#.
#lim_(xrarr0^-) cosx = 1# and #lim_(xrarr0^-) 1= 1#
so #lim_(xrarr0^-) sinx/x = 1#
Since both one sided limits are #1#, the limit is #1#.
Note
This proof uses the fact that #lim_(xrarr0)cosx = 1#. That can also be stated "the cosine function is continuous at #0#".
That fact can be proved from the fact that #lim_(xrarr0) sinx = 0#. (The sine function is continuous at #0#.)
Which can be proved using the squeeze theorem in a argument rather like the one used above.
Furthermore: Using both of those facts we can show that the sine and cosine functions are continuous at every real number.