Answer to Question #205465 in Real Analysis for Aman

Question #205465

Find lim x tends to 0 1-cos2x /x2sinx2


1
Expert's answer
2021-06-11T02:47:02-0400
limx01cos2(x)x2sin(x2)=limx0sin2(x)x2sin(x2)\lim\limits_{x\to 0}\dfrac{1-\cos^2(x)}{x^2\sin(x^2)}=\lim\limits_{x\to 0}\dfrac{\sin^2(x)}{x^2\sin(x^2)}

=limx0((sin(x)x)21sin(x2))=\lim\limits_{x\to 0}\bigg((\dfrac{\sin(x)}{x})^2\cdot\dfrac{1}{\sin(x^2)}\bigg)

=(1)2limx0(1sin(x2))==(1)^2\lim\limits_{x\to 0}(\dfrac{1}{\sin(x^2)})=\infin


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