Advanced High-School Mathematics

SECTION 5.1 Quick Survey of Limits 261

Exercises

1. Using l’Hˆopital’s rule if necessary, compute the limits indicated
   below:

(a) limx→ 1

x^3 − 1

4 x^3 −x− 3

(b) limx→ 1

cos(πx/2)

» (^3) (x−1) 2
(c) limx→∞
2 x^2 − 5 x
x^3 −x+ 10
(d) lim
θ→ 0
sin 3θ
sin 4θ
(e) lim
θ→ 0
sinθ^2
θ
(f) lim
θ→π/ 2
1 −sinθ
1 + cos 2θ
(g) limx→∞
ln(x+ 1)
log 2 x
(h) lim
x→ 0 +
(lnx − ln sinx) (Hint:
you need to convert this
“∞ − ∞” indeterminate
form to one of the forms dis-
cussed above!)
(i) limx→∞(ln 2x−ln(x+ 1)).
(j) limx→∞
(
1 +

4

x

)x

(k) limx→∞

Ç

1 +

a

x

åx

(l) lim

x→ 1

x^1 /(x−1)

(m) limx→∞x^3 e−x

(n) lim

x→ 0 +

xae−x, a > 0

(o) lim

x→ 0 +

lnxln(1−x)

(p) lim

x→ 1 −

lnxln(1−x) (Are (o)

and (p) really different?)

2. Compute

∫∞

0 xe

− 2 xdx.

3. Let n be a non-negative integer. Using mathematical induction,
   show that

∫∞

0 x

ne−xdx=n!.

4. (The (real)∫ Gamma function) Let z > 0 and define Γ(z) =
   ∞
   0 x

z− (^1) e−xdx. Show that
(a) Γ(n) = (n−1)! for any positive integern;
(b) Γ(z) exists (i.e., the improper integral converges) for allz >0.

5. (Convolution) Given functionsf andgdefined for allx∈ R, the
   convolutionoff andgis defined by

f∗g(x) =

∫∞

−∞f(t)g(x−t)dt,