Question

limit as x approaches pi/2 of (x-pi/2)/cos(x)

Answer 1

are you in calculus?

Answer 2

yes. Question is dealing with trig limits without using any derivative work.

Answer 3

oh no derivatives :{ its a 0/0 form so i would use L'hopitals thm (differentiate top/bottom)

Answer 4

*without* using derivatives
so substitute x-pi/2 = y
what u get ?

Answer 5

thats what i wanted to do at first but seeing as my professor wants the limit without derivatives im stuck. I tried using a substitution for x-pi/2 so the function became u/cos(u+pi/2) but couldnt think of what to do next.

Answer 6

simplify cos (u+pi/2) =... ? know what it is ?

Answer 7

sin(u)?

Answer 8

you know cos (A+B) formula ?

Answer 9

no. trig formulas were never really taught while i was in high school so most trig stuff goes over my head.

Answer 10

cos (A+B) =cos A cos B - sin A sin B
put A = u, B= pi/2
what u get ?

Answer 11

-sin(u)

Answer 12

correct!
so your limit will now become ?

Answer 13

0/-1

Answer 14

?
lim u-> 0 -u/ sin u
correct ?
and you know what lim x->0 sin x/x = ... ?

Answer 15

you can just relate sinx/x to -u/sinu? i know the limit of sinx/x as x goes to 0 is 1. so that makes -u/sinu = -1?

Answer 16

absolutely!
and this is how,
lim u-> 0 -u/ sin u = - lim u->0 1/ (sin u/ u) = -1/ [lim u->0 sin u / u] = -1/1
got this ?

Answer 17

ok that makes sense. Thanks for the help

Answer 18

welcome ^_^

Answer 19

and WELCOME to Open Study!! :)