- #1
tylersmith7690
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Homework Statement
evaluate the following limit.
limx→∞ cos3x-cos4x/x^2 , include theorems
Homework Equations
Im guessing its a sandwhich theorem limit and that cos3x-cos4x as an upper bound of 3 and lower of -3. But was wondering if anyone can help explain to me why this is so and should I show a proof or working for this assumption.
The Attempt at a Solution
-3 ≤ cos 3x - cos 4x ≤ 3 , divide through by x^2 to get original equation
-3/x^2 ≤ ( cos 3x - cos 4x ) / x2≤ 4/x2
Now limx→∞ -3/x2 = 0 and
limx→∞ 3/x2 = 0.
So the limx→∞ ( cos 3x - cos 4x ) / x2 = 0 by the squeeze theorem.
I'm not sure as to what rules I would have to put in this working out. Except maybe something
about -1≤ cos ≤ 1 and how cos(3x) = cos(2x + x) = 3cos3-cos x
and cos(4x) = 2 cos2(2x) - 1 = 2(2 cos2(x) - 1)2 - 1 = 8 cos4(x) - 8 cos2(x) + 1
but i don't see how this would help.