Choose the correct simplification of (7x3 − 8x − 5) + (3x3 + 7x + 1).
2 answers:
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To demonstrate a method for computing the limit itself, let's pick a small value of n. If n = 3, then our limit is
Let a = 1 and b the cosine product, and write them as
with
Now we use the identity
to rationalize the numerator. This gives
As x approaches 0, both a and b approach 1, so the polynomial in a and b in the denominator approaches 6, and our original limit reduces to
For the remaining limit, use the Taylor expansion for cos(x) :
where essentially means that all the other terms in the expansion grow as quickly as or faster than x⁴; in other words, the expansion behaves asymptotically like x⁴. As x approaches 0, all these terms go to 0 as well.
Then
so in our limit, the constant terms cancel, and the asymptotic terms go to 0, and we end up with
Unfortunately, this doesn't agree with the limit we want, so n ≠ 3. But you can try applying this method for larger n, or computing a more general result.
Edit: some scratch work suggests the limit is 10 for n = 6.
Answer:
The whale is at a depth 120 feet below the sea level.
Step-by-step explanation:
A grey whale is cruising at a depth of 233 feet below sea level. The depth below the sea level has the negative position, thus the initial position of the whale is -233.
It rises 73 feet, the position of the whale becomes 73 feet higher. This means we should add 73.
Then it comes up another 40 feet, so we need to add 40.
Mathematically,
The whale is at a depth 120 feet below the sea level.