AP Calculus BC Quiz
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LIMITS AND CONTINUITY
Question 1
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Define
r
(
x
)
=
{
sin
x
x
,
x
≠
0
c
,
x
=
0
r(x)=\begin{cases}\dfrac{\sin x}{x},&x\ne0\\c,&x=0\end{cases}
r
(
x
)
=
⎩
⎨
⎧
x
sin
x
,
c
,
x
=
0
x
=
0
; which condition makes
r
r
r
continuous on
[
−
1
,
1
]
[-1,1]
[
−
1
,
1
]
?
A
Set
c
=
0
c=0
c
=
0
so
r
(
0
)
=
0
r(0)=0
r
(
0
)
=
0
and
sin
0
=
0
\sin 0=0
sin
0
=
0
.
B
Set
c
=
1
c=1
c
=
1
so
lim
x
→
0
sin
x
x
=
r
(
0
)
\lim_{x\to0}\dfrac{\sin x}{x}=r(0)
lim
x
→
0
x
sin
x
=
r
(
0
)
and
r
r
r
is defined on
[
−
1
,
1
]
[-1,1]
[
−
1
,
1
]
.
C
Any
c
c
c
works because
sin
x
x
\dfrac{\sin x}{x}
x
sin
x
is continuous for
x
≠
0
x\ne0
x
=
0
.
D
No
c
c
c
works because
sin
x
x
\dfrac{\sin x}{x}
x
sin
x
has an infinite discontinuity at
0
0
0
.
E
Set
c
=
sin
(
1
)
c=\sin(1)
c
=
sin
(
1
)
so endpoint behavior matches at
x
=
1
x=1
x
=
1
.
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