Solving $lim_{xto 0}bigg( Big( frac{1+sin(x)cos(alpha x)}{1+sin(x)cos(beta x)} Big) ^{cot^3(x)}bigg)$: my...











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$$lim_{xto 0}bigg( Big( frac{1+sin(x)cos(alpha x)}{1+sin(x)cos(beta x)} Big) ^{cot^3(x)}bigg)$$
$$lim_{xto 0} bigg( Big( frac{1+sin(x)cos(alpha x)+sin(x)cos(beta x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)} Big) ^{cot^3(x)}bigg)$$



$$lim_{xto 0} bigg( Big( 1+frac{sin(x)cos(alpha x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)}Big)^{cot^3(x) cdot frac{1+sin(x)cos(beta x)}{sin(x)cos(alpha x)-sin(x)cos(beta x)} cdot frac{sin(x)cos(alpha x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)}} bigg) $$



$$e^{lim_{xto 0} big( frac{cos^3(x)sin(x)(cos(alpha x) - cos(beta x))}{sin(x)sin^2(x) (1+sin(x)cos(beta x))} big)}$$



Currently disregarding the exponent, because the font is small.



$$lim_{xto 0} big( frac{cos(alpha x) - cos(beta x)}{sin^2(x)} big)$$



Through Taylor's expansion,



$$lim_{xto 0} big( frac{1-frac{alpha^2x^2}{2} -1 + frac{beta^2x^2}{2}}{x^2} big)$$



And the answer is



$$e^{frac{beta^2 - alpha^2}{2}}.$$



The answer in the book, however, is $e^{beta^2 - alpha^2}$. The formula for the difference of two cosines also leaves me with $e^{frac{beta^2 - alpha^2}{2}}$, and I feel like there's a typo, but maybe I went wrong somewhere?






calculus limits proof-verification







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  • 2




    Your book has a typo.
    – Paramanand Singh
    Nov 18 at 9:11










  • @ParamanandSingh Thanks! Do you suggest I delete this post, answer to myself and accept my own answer, or wait for another user to post the answer so I could accept it? I don't want my question hanging adding to the pile of unanswered ones.
    – fragileradius
    Nov 18 at 9:16










  • Answer yourself and accept it.
    – Paramanand Singh
    Nov 18 at 9:26















up vote
0
down vote

favorite












$$lim_{xto 0}bigg( Big( frac{1+sin(x)cos(alpha x)}{1+sin(x)cos(beta x)} Big) ^{cot^3(x)}bigg)$$
$$lim_{xto 0} bigg( Big( frac{1+sin(x)cos(alpha x)+sin(x)cos(beta x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)} Big) ^{cot^3(x)}bigg)$$



$$lim_{xto 0} bigg( Big( 1+frac{sin(x)cos(alpha x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)}Big)^{cot^3(x) cdot frac{1+sin(x)cos(beta x)}{sin(x)cos(alpha x)-sin(x)cos(beta x)} cdot frac{sin(x)cos(alpha x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)}} bigg) $$



$$e^{lim_{xto 0} big( frac{cos^3(x)sin(x)(cos(alpha x) - cos(beta x))}{sin(x)sin^2(x) (1+sin(x)cos(beta x))} big)}$$



Currently disregarding the exponent, because the font is small.



$$lim_{xto 0} big( frac{cos(alpha x) - cos(beta x)}{sin^2(x)} big)$$



Through Taylor's expansion,



$$lim_{xto 0} big( frac{1-frac{alpha^2x^2}{2} -1 + frac{beta^2x^2}{2}}{x^2} big)$$



And the answer is



$$e^{frac{beta^2 - alpha^2}{2}}.$$



The answer in the book, however, is $e^{beta^2 - alpha^2}$. The formula for the difference of two cosines also leaves me with $e^{frac{beta^2 - alpha^2}{2}}$, and I feel like there's a typo, but maybe I went wrong somewhere?






calculus limits proof-verification







share|cite|improve this question




















  • 2




    Your book has a typo.
    – Paramanand Singh
    Nov 18 at 9:11










  • @ParamanandSingh Thanks! Do you suggest I delete this post, answer to myself and accept my own answer, or wait for another user to post the answer so I could accept it? I don't want my question hanging adding to the pile of unanswered ones.
    – fragileradius
    Nov 18 at 9:16










  • Answer yourself and accept it.
    – Paramanand Singh
    Nov 18 at 9:26













up vote
0
down vote

favorite









up vote
0
down vote

favorite











$$lim_{xto 0}bigg( Big( frac{1+sin(x)cos(alpha x)}{1+sin(x)cos(beta x)} Big) ^{cot^3(x)}bigg)$$
$$lim_{xto 0} bigg( Big( frac{1+sin(x)cos(alpha x)+sin(x)cos(beta x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)} Big) ^{cot^3(x)}bigg)$$



$$lim_{xto 0} bigg( Big( 1+frac{sin(x)cos(alpha x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)}Big)^{cot^3(x) cdot frac{1+sin(x)cos(beta x)}{sin(x)cos(alpha x)-sin(x)cos(beta x)} cdot frac{sin(x)cos(alpha x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)}} bigg) $$



$$e^{lim_{xto 0} big( frac{cos^3(x)sin(x)(cos(alpha x) - cos(beta x))}{sin(x)sin^2(x) (1+sin(x)cos(beta x))} big)}$$



Currently disregarding the exponent, because the font is small.



$$lim_{xto 0} big( frac{cos(alpha x) - cos(beta x)}{sin^2(x)} big)$$



Through Taylor's expansion,



$$lim_{xto 0} big( frac{1-frac{alpha^2x^2}{2} -1 + frac{beta^2x^2}{2}}{x^2} big)$$



And the answer is



$$e^{frac{beta^2 - alpha^2}{2}}.$$



The answer in the book, however, is $e^{beta^2 - alpha^2}$. The formula for the difference of two cosines also leaves me with $e^{frac{beta^2 - alpha^2}{2}}$, and I feel like there's a typo, but maybe I went wrong somewhere?






calculus limits proof-verification







share|cite|improve this question















$$lim_{xto 0}bigg( Big( frac{1+sin(x)cos(alpha x)}{1+sin(x)cos(beta x)} Big) ^{cot^3(x)}bigg)$$
$$lim_{xto 0} bigg( Big( frac{1+sin(x)cos(alpha x)+sin(x)cos(beta x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)} Big) ^{cot^3(x)}bigg)$$



$$lim_{xto 0} bigg( Big( 1+frac{sin(x)cos(alpha x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)}Big)^{cot^3(x) cdot frac{1+sin(x)cos(beta x)}{sin(x)cos(alpha x)-sin(x)cos(beta x)} cdot frac{sin(x)cos(alpha x)-sin(x)cos(beta x)}{1+sin(x)cos(beta x)}} bigg) $$



$$e^{lim_{xto 0} big( frac{cos^3(x)sin(x)(cos(alpha x) - cos(beta x))}{sin(x)sin^2(x) (1+sin(x)cos(beta x))} big)}$$



Currently disregarding the exponent, because the font is small.



$$lim_{xto 0} big( frac{cos(alpha x) - cos(beta x)}{sin^2(x)} big)$$



Through Taylor's expansion,



$$lim_{xto 0} big( frac{1-frac{alpha^2x^2}{2} -1 + frac{beta^2x^2}{2}}{x^2} big)$$



And the answer is



$$e^{frac{beta^2 - alpha^2}{2}}.$$



The answer in the book, however, is $e^{beta^2 - alpha^2}$. The formula for the difference of two cosines also leaves me with $e^{frac{beta^2 - alpha^2}{2}}$, and I feel like there's a typo, but maybe I went wrong somewhere?






calculus limits proof-verification




calculus limits proof-verification






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edited Nov 18 at 9:22









Kemono Chen

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asked Nov 18 at 9:08









fragileradius

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  • 2




    Your book has a typo.
    – Paramanand Singh
    Nov 18 at 9:11










  • @ParamanandSingh Thanks! Do you suggest I delete this post, answer to myself and accept my own answer, or wait for another user to post the answer so I could accept it? I don't want my question hanging adding to the pile of unanswered ones.
    – fragileradius
    Nov 18 at 9:16










  • Answer yourself and accept it.
    – Paramanand Singh
    Nov 18 at 9:26














  • 2




    Your book has a typo.
    – Paramanand Singh
    Nov 18 at 9:11










  • @ParamanandSingh Thanks! Do you suggest I delete this post, answer to myself and accept my own answer, or wait for another user to post the answer so I could accept it? I don't want my question hanging adding to the pile of unanswered ones.
    – fragileradius
    Nov 18 at 9:16










  • Answer yourself and accept it.
    – Paramanand Singh
    Nov 18 at 9:26








2




2




Your book has a typo.
– Paramanand Singh
Nov 18 at 9:11




Your book has a typo.
– Paramanand Singh
Nov 18 at 9:11












@ParamanandSingh Thanks! Do you suggest I delete this post, answer to myself and accept my own answer, or wait for another user to post the answer so I could accept it? I don't want my question hanging adding to the pile of unanswered ones.
– fragileradius
Nov 18 at 9:16




@ParamanandSingh Thanks! Do you suggest I delete this post, answer to myself and accept my own answer, or wait for another user to post the answer so I could accept it? I don't want my question hanging adding to the pile of unanswered ones.
– fragileradius
Nov 18 at 9:16












Answer yourself and accept it.
– Paramanand Singh
Nov 18 at 9:26




Answer yourself and accept it.
– Paramanand Singh
Nov 18 at 9:26










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      There's a typo, as affirmed in the comments.






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        answered Nov 18 at 9:28









        fragileradius

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