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Unformatted text preview: Solutions to Bonus Quiz B
www.math.uﬂ.edu/˜harringt
February 17, 2008
1. Find all the vertical asymptotes for f (x) =
Note that f (x) = (x+1) sin(x)
x3 − x = x+1
x+1 ∗ sin x
x ∗ (x+1) sin(x)
.
x3 − x 1
.
x− 1 So we have holes at x = −1 and at x = 0 since limx→0
x+1
limx→−1 x+1 = 1. sin x
x = 1 and Thus, we have ONE vertical asymptote at x = 1.
2. Solve for x: e3x = 6e4x+1
Note: There are multiple ways to do this problem. I will demonstrate
only two. e3x =
e3x − 6e4x+1 =
e3x − 6e3x+x+1 =
e3x (1 − 6ex+1 ) =
1 − 6ex+1 =
1=
1/6 =
ln 1/6 − 1 = 6e4x+1
0
0
0 Note: that e3x = 0 for all x
0 We divided both sides by e3x
6ex+1
ex+1
x Or....
1 e3x
ln(e3 x)
3x
3x
−x
x 6e4x+1
ln(6 ∗ e4x+1 )
ln(6) + ln(e4x+1 )
ln(6) + 4x + 1
ln(6) + 1
− ln(6) − 1 =
=
=
=
=
= 3. Evaluate the following limit: limx→0 tan x
x tan x
sin(x)
sin(x)
1
= lim
= lim
∗ lim
=1∗1=1
x→0
x→0 x cos(x)
x→0
x→0 cos x
x
x
lim 4. Determine which of the following is even: x sin x or
answer. x
.
x Explain your Recall that a function is even when f (−x) = f (x).
(−x) sin(−x) = −x ∗ (− sin(x)) = x ∗ sin x Thus, x sin x is EVEN.
−x
−x = x
−x = − x Hence,
x x
x is ODD. 5. Compute the following limits. Let
√ 9 − x for x < 0
f (x) =
x2 + 3 for 0 ≤ x < 1
3
x − x for x > 1
(a) limx→0 f (x) = 3
√
Since limx→0− f (x) = 9 − 0 = 3 and limx→0+ f (x) = 02 + 3 = 3
(b) limx→1− f (x) = 12 + 3 = 4
(c) limx→0+ f (x) = 02 + 3 = 3
(d) limx→1 f (x) =Does Not Exist
Since limx→1− f (x) = 12 + 4 = 4 and limx→1+ f (x) = 13 − 1 = 3 2 ...
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 Fall '08
 ALL
 Calculus, Geometry, Asymptotes

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