Calculus
posted by John on .
How would you find the limit of (secx1)/(x^2) as x goes to 0 algebraically?

change secx 1 to 1/cosx 1 to (1cosx)/cosx
then
lim (1cosx)/(cosx)x^2 rationalize the numberator.
lim (1cosx)(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
(secx1)(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 (secx1)/(x^2) = 1/2
Alternatively, use l'Hôpital's rule.