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wu :: forums    riddles    putnam exam (pure math) (Moderators: william wu, towr, SMQ, Icarus, Grimbal, Eigenray)    Probably easy limit question « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Probably easy limit question  (Read 2275 times) Lizzy Newbie     Posts: 5 Probably easy limit question   « on: Mar 20th, 2006, 7:00pm » Quote Modify Hey guys, this will probably be really easy for u...but i don't know how to do it   Find the limit of   cot(2x)sin(6x) as x approaches 0   any help on how to do it will be greatly appreciated   Lizzy   xoxo IP Logged Barukh Uberpuzzler     Gender: Posts: 2276 Re: Probably easy limit question   « Reply #1 on: Mar 20th, 2006, 11:41pm » Quote Modify Making a substitution z = 2x, you simplify the limit to cot(z)sin(3z), so actually you need to find the limit of sin(3z)/sin(z). Use the trigonometric identity for a triple angle to obtain the answer 3.   Another way is to look at the power series for cot(x) and sin(x). IP Logged Icarus wu::riddles Moderator Uberpuzzler Boldly going where even angels fear to tread.     Gender: Posts: 4863 Re: Probably easy limit question   « Reply #2 on: Mar 21st, 2006, 3:45pm » Quote Modify Another approach: recall the identities limz->0 (sin z)/z = 1 and limz->0 (tan z)/z = 1.   The second can be rewritten as limz->0 z cot z = 1. So,   lim cot 2x sin 6x = lim 3(2x cot 2x)((sin 6x)/6x)    = 3 (lim 2x cot 2x)(lim (sin 6x)/6x) = 3 * 1 * 1 = 3. IP Logged "Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous Lizzy Newbie     Posts: 5 Re: Probably easy limit question   « Reply #3 on: Mar 22nd, 2006, 2:29am » Quote Modify yay! i understand icarus' method   hmm, does anyone know any good online resources to learn this stuff properly? IP Logged Icarus wu::riddles Moderator Uberpuzzler Boldly going where even angels fear to tread.     Gender: Posts: 4863 Re: Probably easy limit question   « Reply #4 on: Mar 22nd, 2006, 6:19pm » Quote Modify Sorry, but I don't. I learned it before the internet was easily accessible. IP Logged "Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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