Question

证明当x趋向0时,有1.arctanx~x2.secx-1~1/2x^2求答案

Answer 1

x→0

lim arctanx / x

换元t=arctanx

=lim(t→0) t / tant

=lim (cost)*(t / sint)

=1

lim (secx-1) / (x^2/2)

=lim (1-cosx)/cosx / (x^2/2)

=lim (2sin^2(x/2)) / (x^2/2) * (1/cosx)

=lim [sin(x/2) / (x/2)]^2 * (1/cosx)

=1

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