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Math Help - need help plz infinite limit

  1. #1
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    need help plz infinite limit

    Let f(x)= x e^(-x)
    (a) determine the domain of the function f(x)
    (b)Is f(x) an odd or even orneither
    (c)determine any discontunity point for f(x)
    (d)evaluate lim (x->infinity) f(x) and lim (x->-infinity) f(x)
    e)find critical numbers and where the function is increasing or decreasing
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by gracy View Post
    Let f(x)= x e^(-x)
    (a) determine the domain of the function f(x)
    f(x) is defined for all real x, so its domain is R

    (b)Is f(x) an odd or even orneither
    f(-x)=(-x)e^(x)

    which is not equal to either f(x), or -f(x), so f is neither odd nor even

    (c)determine any discontunity point for f(x)
    neither x nor e^(-x) has a discontinuity in R so neither does their
    product.

    (d)evaluate lim (x->infinity) f(x) and lim (x->-infinity) f(x)
    L'hopital's rule is needed here:

    f(x)=x e(-x) = x/e(x)

    so:

    lim(x->inf) f(x) = (lim(x->inf) d/dx(x) )/(lim(x->inf) d/dx(e^x)) =1/(lim(x->inf) e^x)=0

    and

    lim(x->-inf) f(x) = (lim(x->-inf) d/dx(x) )/(lim(x->-inf) d/dx(e^x)) =1/(lim(x->-inf) e^x)=inf

    e)find critical numbers and where the function is increasing or decreasing
    (d/dx) x e^(-x)=e^(-x) + x (-1) e^(-x)=e^(-x)[1-x]

    thus the critical points are the solutions of:

    e^(-x)[1-x]=0,

    but e^(-x) is never zero., so we want the solution/s of 1-x=0, which is
    of course x=1.

    For x large positive df/dt is negative, so f is decreasing on (1,inf), and so
    increasing on (-inf,1).

    RonL
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