Thread: 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

Originally Posted by gracy

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