Answer to Question #186449 in Real Analysis for Harshita

Question #186449

Find the following limit

Lim x tengs to 0 1-cos x^2/x^2 sin x^2


1
Expert's answer
2021-05-07T09:13:04-0400

"Solution:~ Let~L=\\lim_{x\\to 0} \\frac{1- cos~ x^2 }{x^2 ~sin ~x^2}\n\\\\This ~is ~ indeterminant ~form.Therefore~we~apply~L'Hospital ~Rule ~to ~Solve~this ~limit.\n\\\\\\therefore Apply~L'Hospitals ~Rule\n\\\\\\therefore~L= \\lim_{x\\to 0} \\frac{2x.sin~ x^2 }{2x.sin~ x^2+2x^3 ~cos ~x^2}\n\\\\~~~~~~~~= \\lim_{x\\to 0} \\frac{2x.sin~ x^2 }{2x(sin~ x^2+x^2 ~cos ~x^2)}\n\\\\~~~~~~~~= \\lim_{x\\to 0} \\frac{sin~ x^2 }{sin~ x^2+x^2 ~cos ~x^2}\n\\\\ which ~is ~again ~ indeterminant ~form. Therefore~we~again~apply ~L'Hospital ~Rule.\n\\\\\\therefore L=\\lim_{x\\to 0} \\frac{2x ~cos~ x^2 }{4x~cos~ x^2-2x^3 ~sin ~x^2}\n\\\\~~~~~~~~=\\lim_{x\\to 0} \\frac{2x ~cos~ x^2 }{2x~(2~cos~ x^2-x^2 ~sin ~x^2)}\n\\\\~~~~~~~~=\\lim_{x\\to 0} \\frac{cos~ x^2 }{2~cos~ x^2-x^2 ~sin ~x^2}\n\\\\Now ~put ~the ~limit ~x=0\n\\\\\\therefore L=\\lim_{x\\to 0} \\frac{cos~ 0^2 }{2~cos~ 0^2-0^2 ~sin ~0^2}=\\frac{1}{2}\n\\\\\\lim_{x\\to 0} \\frac{1- cos~ x^2 }{x^2 ~sin ~x^2}=\\frac{1}{2}"


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