Posted by John on Sunday, September 26, 2010 at 8:22pm.
change secx -1 to 1/cosx -1 to (1-cosx)/cosx
then
lim (1-cosx)/(cosx)x^2 rationalize the numberator.
lim (1-cosx)(1+cosx)/(1+cosx)cosx x^2
lim (sin^2x)/x^2 * 1/(1+cosx)cosx
lim (sinx/x)lim sinx/x Lim (1/(cosx)(1+cosx)
1*1*1/(1*2)= 1/2
Multiply top and bottom by (sec(x)+1) to get
(secx-1)(sec(x)+1)/((x^2)(sec(x)+1)
=(sec²(x)-1)/((x^2)(sec(x)+1)
=(sin(x)/x)²/(cos²(x)(1+sec(x)))
Lim x->0 (sin(x)/x)=1
Lim x->0 cos(x)=1
Lim x->0 sec(x)=1
Therefore
Lim x->0 (secx-1)/(x^2) = 1/2
Alternatively, use l'Hôpital's rule.
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