SME
Calculus I · Problem type
The limit (1 + a ÷ x)ᵇˣ
Use this for “Find the limit of (1 + 2 ÷ x)³ˣ as x → ∞” (and of (1 + 2x)3 ÷ x as x → 0, the same limit).
Chart
f(1,000) ≈ 401
Level L = 403.4 never meets the curve
This is an example. Type your own values to start a new problem.
lim
x
→
∞
(1
+
/x)
x
=
e
=
L
x
Large x to test
example
y
Value of y at that x
worked out (not typed)
Formulas
ln
L
=
a
×
b
6 = 2 × 3
L
=
e
ln
L
403.4288
=
e
6
y
=
(1 +
a
x
)
b
×
x
401.0187
=
(1 +
2
1,000
)
3 × 1,000
Assumptions
•
As x → ∞ the base 1 + a ÷ x → 1 while the power bx grows without bound. That form is indeterminate, so take ln first.
•
ln y = bx·ln(1 + a ÷ x) = ln(1 + a ÷ x) ÷ (1 ÷ (bx)), a 0/0 form that L’Hôpital’s rule can take.
•
Differentiate the top and the bottom separately (not the quotient rule). Their ratio ab ÷ (1 + a ÷ x) tends to ab.
•
Undo the ln at the end: L = eᵃᵇ. The test x must keep the base 1 + a ÷ x above 0.
Step-by-step
WE KNOW
a = 2, b = 3, x = 1,000
FIND
limit of ln y (ln L), limit of y (L), value of y at that x (y)
Step 1 · Find limit of ln y (ln L)
Use ln L = a × b
Write
ln
y
as
ln(1 +
a
÷
x
)
1 ÷ (
bx
)
,
both
parts
→
0,
and
take
the
slope
of
each:
their
ratio
tends
to
ab.
ln
L
=
2
×
3
f
(
x
)
=
ln(1
+
2
x
)
→
f
′(
x
)
=
−2 ÷
x
2
1 + 2 ÷
x
g
(
x
)
=
1
3
x
→
g
′(
x
)
=
−
1
3
x
²
lim
x
→
∞
of
ln(1 + 2 ÷
x
)
1 ÷ (3
x
)
=
6
ln
L
=
6
Step 2 · Find limit of y (L)
Use L = eˡⁿᴸ
ln
y
→
ab,
so
y
itself
→
e
to
that
power:
undo
the
ln.
L
=
e
6
L
=
403.4288
Step 3 · Find value of y at that x (y)
Use y = (1 + a ÷ x)b × x
Put
the
test
x
into
y
to
check:
the
larger
x
is,
the
closer
y
comes
to
L
.
y
=
(1 +
2
1,000
)
3 × 1,000
y
=
(1 + 0.002)
3,000
y
=
1.002
3,000
y
=
401.0187
Check
Put the numbers back in. Both sides should match.
6 = 2 × 3 ✓
403.4288
=
e
6
✓
401.0187
=
(1 +
2
1,000
)
3 × 1,000
✓
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