Proof of: limq?0 (cosq -1)/q = 0 trig limit, Mathematics

Proof of: limq→0 (cosq -1)/q = 0

We will begin by doing the following,

limq→0 (cosq -1)/q = limq→0((cosq - 1)(cosq + 1))/(q (cosq + 1))

= limq→0(cos2q - 1)/ (q (cosq + 1))                                           (7)

Here, let's recall that,

cos2q + sin2q = 1      =>   cos2q - 1 = -sin2q

By using this in (7) provides us,

= limq→0(sin2q)/ (q (cosq + 1))

= limq→0 (sinq/q)(-sinq)/(cosq + 1)

= limq→0 (sinq/q)  limq→0 (-sinq)/(cosq + 1)

Here, as we just proved the first limit and the second can be got directly we are pretty much done.  All we require to do is get the limits.

limq→0 cosq - 1

= limq→0 (sinq/q)  limq→0 (-sinq)/(cosq + 1)

= (1) (0)

 = 0

Posted Date: 4/13/2013 3:52:18 AM | Location : United States

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