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Math Help - Another hard limit

  1. #1
    Senior Member polymerase's Avatar
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    Another hard limit

    FInd the value of \lim_{x\to\frac{\pi}{4}} \frac{tan\:2x}{cot(\frac{\pi}{4}-x)}

    Thanks!
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  2. #2
    Super Member PaulRS's Avatar
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    Let u=\frac{\pi}{4}-x

    L=\lim_{x\rightarrow{\frac{\pi}{4}}}\frac{\tan(2x)  }{\cot(\frac{\pi}{4}-x)}=\lim_{u\rightarrow{0}}\frac{\tan(2(\frac{\pi}{  4}-u))}{\cot(u)}

    \tan(2x)=\frac{2\tan(x)}{1-\tan^2(x)}

    L=\lim_{u\rightarrow{0}}\frac{\tan(2(\frac{\pi}{4}-u))}{\cot(u)}=\lim_{u\rightarrow{0}}\frac{\frac{2\  tan(\frac{\pi}{4}-u)}{1-\tan^2(\frac{\pi}{4}-u)}}{\cot(u)}

    L=\lim_{u\rightarrow{0}}\frac{\frac{2\tan(\frac{\p  i}{4}-u)}{1-\tan^2(\frac{\pi}{4}-u)}}{\cot(u)}=\lim_{u\rightarrow{0}}\frac{\frac{2\  tan(\frac{\pi}{4}-u)}{1-\tan^2(\frac{\pi}{4}-u)}}{\frac{\cos(u)}{\sin(u)}}=2\lim_{u\rightarrow{  0}}\frac{\sin(u)}{1-\tan^2(\frac{\pi}{4}-u)}

    L=2\lim_{u\rightarrow{0}}\frac{\sin(u)}{1-\tan^2(\frac{\pi}{4}-u)}=2\lim_{u\rightarrow{0}}\frac{\sin(u)}{(1-\tan(\frac{\pi}{4}-u))(1+\tan(\frac{\pi}{4}-u))}

    L=\lim_{u\rightarrow{0}}\frac{\sin(u)}{1-\tan(\frac{\pi}{4}-u)}=\frac{\sqrt[]{2}}{2}\lim_{u\rightarrow{0}}\frac{\sin(u)}{\cos(\  frac{\pi}{4}-u)-\sin(\frac{\pi}{4}-u)}

    Now recall the difference formulas: \cos(\frac{\pi}{4}-u)=\frac{\sqrt[]{2}}{2}(\cos(u)+\sin(u))

    And: \sin(\frac{\pi}{4}-u)=\frac{\sqrt[]{2}}{2}(\cos(u)-\sin(u))

    Thus: L=\frac{\sqrt[]{2}}{2}\lim_{u\rightarrow{0}}\frac{\sin(u)}{\frac{  \sqrt[]{2}}{2}(2\sin(u))}\lim_{u\rightarrow{0}}\frac{\sin  (u)}{2\sin(u)}=\frac{1}{2}
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  3. #3
    Super Member wingless's Avatar
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    Do you know l'Hôpital's rule? If your limit calculations give you \frac {0}{0} or \frac {\infty}{\infty}, you can differentiate the numerator and denominator to get a result. In this question, you have to differentiate them a few times, just keep doing it until you get a result except \frac {0}{0} or \frac {\infty}{\infty}.
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  4. #4
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    Quote Originally Posted by wingless View Post
    Do you know l'Hôpital's rule?
    Do you know that L'Hôpital's Rule does not belong to Guillaume de L'Hôpital?

    Do you know that it's more interesting solvin' limits without such Rule?

    --

    L'Hôpital's Rule is boring! I'd use it when extremely be necessary. (For a really, really, really hard limit.)
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  5. #5
    Super Member wingless's Avatar
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    What would you like me to call the rule? It's not called "Bernoulli's Rule", it's called "L'Hôpital's Rule"..

    I just gave him a way to solve the problem. You have the choice to use it or not.
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  6. #6
    Senior Member polymerase's Avatar
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    Quote Originally Posted by Krizalid View Post
    Do you know that L'Hôpital's Rule does not belong to Guillaume de L'Hôpital?

    Do you know that it's more interesting solvin' limits without such Rule?

    --

    L'Hôpital's Rule is boring! I'd use it when extremely be necessary. (For a really, really, really hard limit.)
    Can you list some REALLY REALLY REALLY hard limits. I want to see
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  7. #7
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by polymerase View Post
    Can you list some REALLY REALLY REALLY hard limits. I want to see
    Never never ask Krizalid to come up with hard problems. You will get what you asked for!

    -Dan
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  8. #8
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    Quote Originally Posted by wingless View Post
    Do you know l'Hôpital's rule?
    Do you know that L'Hopital's rule is for little girls who are afraid to calculate limits? Because when you solve without this rule you got to come up with bound and approximates for the function, which is what a large portion of analysis is about. Thus, solving without L'Hopital teaches you these things.
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  9. #9
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    Quote Originally Posted by PaulRS View Post
    Let u=\frac{\pi}{4}-x

    L=\lim_{x\rightarrow{\frac{\pi}{4}}}\frac{\tan(2x)  }{\cot(\frac{\pi}{4}-x)}=\lim_{u\rightarrow{0}}\frac{\tan(2(\frac{\pi}{  4}-u))}{\cot(u)}
    Just write,
    \lim_{u\to 0}\frac{\tan 2\left( \frac{\pi}{4} - u\right)}{\cot u} = \lim_{u\to 0}\frac{\tan \left( \frac{\pi}{2} - 2u \right)}{\cot u} = \lim_{u\to 0}\frac{\cot 2u}{\cot u} = \lim_{u\to 0}\frac{\tan u}{\tan 2u} = \frac{1}{2}

    Because,
    \frac{\tan u}{\tan 2u} = \frac{\sin u}{\sin 2u} \cdot \frac{\cos 2u}{\cos u}
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  10. #10
    Senior Member polymerase's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Do you know that L'Hopital's rule is for little girls who are afraid to calculate limits? Because when you solve without this rule you got to come up with bound and approximates for the function, which is what a large portion of analysis is about. Thus, solving without L'Hopital teaches you these things.
    I’m just a student...I’m not a mathematician like you guys...I just care about knowing how to solve a question(while learning why) but if I can use l'Hopital's rule and it takes me 1 min. vs 10 min. doing it the "pro" way...I’ll take the little girls' way. As long as I can solve the problem and get the marks...I’m good.
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  11. #11
    Senior Member polymerase's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Just write,
    \lim_{u\to 0}\frac{\tan 2\left( \frac{\pi}{4} - u\right)}{\cot u} = \lim_{u\to 0}\frac{\tan \left( \frac{\pi}{2} - 2u \right)}{\cot u} = \lim_{u\to 0}\frac{\cot 2u}{\cot u} = \lim_{u\to 0}\frac{\tan u}{\tan 2u} = \frac{1}{2}

    Because,
    \frac{\tan u}{\tan 2u} = \frac{\sin u}{\sin 2u} \cdot \frac{\cos 2u}{\cos u}
    ...i did do that but then i got stuck when i got to \frac{tan (\frac{\pi}{2}-2u)}{cot\:u}...i didnt know you could just simplfy to cot\:2u...I knew you could simplfy tan(\frac{\pi}{2}-x) to cot\:x...i guess i dont know my trig well enough
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  12. #12
    Super Member wingless's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Do you know that L'Hopital's rule is for little girls who are afraid to calculate limits? Because when you solve without this rule you got to come up with bound and approximates for the function, which is what a large portion of analysis is about. Thus, solving without L'Hopital teaches you these things.
    That means you have a lot of time to solve every limit without the rule, I don't. Most of the students don't have much time either. I know, this rule 'simplifies' the calculations and makes them less significant. But please don't say "L'Hopital is for noobs", "only little girls use it" etc. People can choose their way to go, if they know the advantages and disadvantages of them.
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