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L'hospital rule

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Evaluate the limit.

lim (cos(x))^(7/x^2) as x goes x-->0^+

  • L'hospital rule - ,

    L'Hôpital's rule applies when there is a fraction whose numerator and denominator are both undefined or zero.

    Since the given expression is not a fraction, we need to transform it to a form where L'Hôpital's rule applies.

    Taking log is a good way when powers are involved:
    ln((cos(x))^(7/x^2))
    =(7/x²)*ln(cos(x))
    =7ln(cos(x))/x²
    Now both numerator and denominator become zero as x->x+.

    differentiate with respect to x:
    7(-sin(x)/cos(x)) / 2x
    =-7tan(x)/2x

    Since the expression is still undefined when x->0+, we can differentiate again:
    -7sec²(x)/2
    =-7/2 as x->0+

    So the original limit can be found by raising -7/2 to the power of e, or
    Lim x->0+ cos(x)^(7/x²)
    = e-7/2

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