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Find the limit as (x,y)->(0,0) of [cos(xy)-1] / (x^2Y^2?)?

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Dr. Juanita Wolff

Feb 20, 2025

lim((x,y)→(0,0)) (cos(xy) – 1)/(x^2 y^2) = lim(t→0) (cos(t) – 1)/t^2, letting t = xy= lim(t→0) (cos(t) – 1)(cos(t) + 1) / [t^2(cos(t) + 1)]= lim(t→0) (cos^2(t) – 1) / [t^2(cos(t) + 1)]= lim(t→0) -sin^2(t) / [t^2(cos(t) + 1)]= lim(t→0) (sin(t)/t)^2 * [-1/(1 + cos t)]= 1^2 * -1/2= -1/2.I hope this helps!... Show More
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Dr. Juanita Wolff

Feb 20, 2025

The limit is -1/2. If you take x = r cosΘ and y = r sinΘ, then for r near zero, cos(xy) = 1 – ½(r^4 cos²Θ sin²Θ) + 1/24 r^8 cos^4Θ sin^4Θ + …so that (cos(xy) – 1)/(x²y²) = – 1/2 + O(r^4)(That is standard big-oh notation on the right.)... Show More
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