Find the limit \(\displaystyle\lim_{{{x}\to{\frac{{\pi}}{{{2}}}}}}{\left({\frac{{{\cos{{\left({5}{x}\right)}}}}}{{{\cos{{\left({3}{x}\right)}}}}}}\right)}\) without using L'Hospital's

Cash Duncan

Cash Duncan

Answered question

2022-04-14

Find the limit limxπ2(cos(5x)cos(3x)) without using L'Hospital's rule

Answer & Explanation

Wernbergbo9d

Wernbergbo9d

Beginner2022-04-15Added 12 answers

cos(5x)=sin(52π5x)=sin5(π2x)
And
cos(3x)=sin(32π3x)=sin3(π2x)
So we set π2x=w
as xπ2 we have x0
The given limit can be written as
limw0sin5wsin3w=53limw03wsin5w5wsin3w=53limw0(sin5w5w3wsin3w)=53
Hope this can be useful
resacarno4u

resacarno4u

Beginner2022-04-16Added 12 answers

Write t=xπ2, then 
limxπ2cos(5x)cos(3x)=limt0sin(5t2π)sin(3tπ)
=limt0sin(5t)sin(3t)
=limt0sin(5t)5t3tsin(3t)53
=53limt0sin(5t)5tlimt03tsin(3t)
=-53·limt0sin(5t)5t=1·(limt0sin(3t)3t=1)-1
=53

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