Calculus

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How would you find the limit of (secx-1)/(x^2) as x goes to 0 algebraically?

  • Calculus -

    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

  • Calculus -

    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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