How do you determine the limit of #(pi/2)-(x)/(cos(x))# as x approaches pi/2?

Question

4831 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-05-20T16:12:32

The limit does not exist, but

As #xrarrpi/2#, #cosxrarr0#, so the expression #x/cosx# either increases or decreases without bound depending on the direction of approach.

However
If the intended question is for the limit of#(pi/2-x)/cosx#, then use the trigonometric identity

#cosx = sin(pi/2-x)# to get

#lim_(xrarrpi/2) (pi/2-x)/cosx = lim_(xrarrpi/2) (pi/2-x)/sin(pi/2-x)#.

With #u = pi/2-x# we have

#lim_(urarr0)u/sinu = 1# by a fundamental trigonometric limit.

That is, #lim_(xrarrpi/2) (pi/2-x)/cosx = 1#.