SME
Calculus I · Problem type
The limit of sin(kx) ÷ (mx)
Use this for “Find the limit of sin(3x) ÷ (2x) as x → 0” or “Find k so that sin(kx) ÷ (2x) → 3 as x → 0.”
Table
An x close to 0 (radians) (x)
sin(kx) ÷ (mx) there (r)
The limit L = 1.5
This is an example. Type your own values to start a new problem.
lim
x
→
0
sin(
x)
x
=
L
x
An x close to 0 (radians)
example
r
sin(kx) ÷ (mx) there
worked out (not typed)
Formulas
L
=
k
m
1.5
=
3
2
r
=
sin(
k
×
x
)
m
×
x
1.4776
=
sin(3 × 0.1)
2 × 0.1
Assumptions
•
x is in radians.
•
sin θ ÷ θ → 1 as θ → 0 (squeeze: cos θ ≤ sin θ ÷ θ ≤ 1 near 0).
•
sin(kx) ÷ (mx) = (k ÷ m) × sin(kx) ÷ (kx), so the limit is k ÷ m.
Step-by-step
WE KNOW
k = 3, m = 2, x = 0.1
FIND
the limit (L), sin(kx) ÷ (mx) there (r)
Step 1 · Find the limit (L)
Use L = k ÷ m
Write
it
as
(
k
m
)
×
sin(
kx
)
kx
;
the
second
factor
→
1.
L
=
3
2
L
=
1.5
Step 2 · Find sin(kx) ÷ (mx) there (r)
Use r = sin(k × x) ÷ (m × x)
Work
the
sine
in
radians,
then
divide.
The
closer
x
is
to
0,
the
closer
r
is
to
k
m
.
r
=
sin(3 × 0.1)
2 × 0.1
r
=
1.4776
Check
Put the numbers back in. Both sides should match.
1.5
=
3
2
✓
1.4776
=
sin(3 × 0.1)
2 × 0.1
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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