356 Chapter 6 • Differentiable Functions
- Prove that \:In E N, lim t: = 0, and use this result to prove that
t-tco e
. e-1/u -l/x^2
(a) \:In EN, hm --= 0 (b) \:In EN, lim _e __ = 0
u-tO+ un X-tO xn
(c) For any polynomialp(x), lim p (l) e-^1 /x = 0 and limp ( l) e-^1 /x
2
= 0.
X-tO+ x X-tO x{'e-l/x if X > 0 }- Define the function f(x) =. '.
0 ifx:'SO
(a) Prove that f has derivatives of every order at every x i= 0. [Show
that, in fact, \:In EN, 3 a polynomial q(x) with constant term 0 such that
J(n)(x) = q (~) e-1/x.]
(b) P rove that \:In E N , j(n) (0) = 0. [Thus, f has derivatives of all orders
everywhere.]
(c) Find the nth Taylor polynomial Tn(x) of f about 0. Prove that
\:Ix E JR, lim Tn(x) exists but is never equal to f(x) when x > 0.
n-tco
16. Define the function f(x) =. '.
{e-l/x
2
if x i= 0 }
0 if x = 0
(a) Prove that f has derivatives of every order at every xi= 0. [See hint
for Exercise 15 .]
(b) Prove that \:In EN, J(n)(O) = 0. [Thus, f has derivatives of all orders
everywhere.]
(c) Find the nth Taylor polynomial Tn(x) of f about 0. Prove that
\:Ix E JR, n-tco lim Tn(x) exists but is never equal to f(x) when xi= 0.17. Use L'Hopital's rule to find each of the following:(a) lim (l+x)^1 /x (b) lim (l+x)l/x_e
X-tCO X-t0 X( c) lim x [ ( 1 t ~ \x) :-e l
X-tCO 1 + X
For (b) and (c), Exercise 6.2.15 will be helpful.