bonusBsol - Solutions to Bonus Quiz B...

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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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bonusBsol - Solutions to Bonus Quiz B...

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