SOLUTION: limit x (1 - 2 cos ((pi x)/(3 x - sqrt3))) as x \[LongRightArrow] + \[Infinity]

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Question 1209595: limit x (1 - 2 cos ((pi x)/(3 x - sqrt3))) as x \[LongRightArrow] + \[Infinity]
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959) About Me  (Show Source):
You can put this solution on YOUR website!
Here's how to evaluate the limit:
**1. Simplify the argument of the cosine function:**
As x approaches infinity, the term -√3 in the denominator becomes insignificant compared to 3x. So, we can approximate the argument of the cosine as:
(πx) / (3x - √3) ≈ (πx) / (3x) = π/3
**2. Evaluate the limit:**
Now, substitute this approximation back into the original expression:
lim (x→∞) [1 - 2cos(πx / (3x - √3))] = 1 - 2cos(π/3)
Since cos(π/3) = 1/2, we have:
1 - 2 * (1/2) = 1 - 1 = 0
Therefore, the limit is 0.

Answer by ikleyn(52750) About Me  (Show Source):
You can put this solution on YOUR website!
.


In his post, @CPhill solved another problem, different from the given in the post.

For correct solution (which is very long), see the AI solution under this link

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