Find the limit as (x,y)->(0,0) of [cos(xy)-1] / (x^2Y^2?)?
I am trying to find the limit of the expression (\frac{\cos(xy) - 1}{x^2y^2}) as ((x,y)) approaches ((0,0)). Can anyone help me with this? Thank you!
2 Answers
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!
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