The problem was to find the limit:
lim x e-x
x → ∞
There are two ways you are likely to have tried L'Hopital's Rule
on this one. One is to observe that
and then set it up as
which is the approach that leads to a solution. If instead you
tried
you undoubtedly found that applying L'Hopital's rule to the above
doesn't seem to help you toward a solution (even though both numerator
and denominator do go to zero as
x goes to infinity). You should
know that very often on these L'Hopital problems, there are two possible
ways you can go, but only on leads easily to a solution. Sometimes you
just have to try them both.
So let's pursue the
setup. Clearly both numerator and denominator go to infinity as
x
goes to infinity. So this expression is a candidate for applying L'Hopital's
rule. We take the derivatives of both the numerator and denominator and
make the new quotient:
lim
x → ∞
|
x
=
ex
|
lim
x → ∞
|
1
=
ex
|
lim e-x = 0
x → ∞
|
which completes the solution.
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