Determine the following limits.
lim x→∞ (5 + (cos^4...

Question

Determine the following limits.

lim x→∞ (5 + (cos⁴ x) / (x² + x + 1))

Accepted Answer

Welcome back, everyone. Evaluate the limit. Limit as X approaches infinity of 9 plus cosine squad of x divided by X2 + 2 X + 4, and we are given 4 answer choices AS 19 divided by 2, B 28 divided by 3, C9, and D 10. So let's analyze this limit. First of all, let's rewrite it. We have limit as x approaches infinity of 9, and we can write the second function as a separate limit based on the properties of limits that's limit as x approaches infinity of cosine squared of x divided by. X2 + 2 x + 4. We can say that the first limit is going to be equal to 9 because it's. The limit of a constant limit of 9 as X approaches any number is going to be equal to 9 because we have a constant function. Now for the second function, we have cosine of X, which oscillates between 1 and 1, right? And for that purpose, we're going to apply the squeeze theorem. We know that. Cosine has a range between -1 and 1 inclusive. So if we analyze cosine square of X, it is going to oscillate between 0 and 1 inclusive because a square of a negative number is going to be non-negative, right? And also, what we're going to do is basically multiply both sides or basically multiply the whole inequality by 1 divided by X2 + 2 X + 4. So, first of all, we're multiplying 0 by. 1 divided by x2 + 2 x + 4, and this still gives us 0, which is going to be less than or equal to cosine square x, we are dividing it by X2 + 2 x + 4, and then for 1, well, we have to divide 1 by X2 + 2 x + 4. And now let's apply the squeeze theorem. We want to evaluate the limit of 0. Limit as x approaches infinity of 0 is basically 0 because we have a constant function and then we want to find the limit as x approaches infinity of 1 divided by x2 + 2 x + 4. If we apply the right substitution, well, we're going to get 1 divided by infinity square plus 2 multiplied by infinity plus 4, which can be combined into infinity, and 1 divided by infinity is going to. Have a limit of 0 because we're dividing 1 by an infinitely large number. So based on the squeeze theorem, because our left hand side limit is equal to 0 and our right hand side limit is equal to 0 as well, then the limit of the middle function is going to be 0 as well, meaning, If we go back to our original limit, this limit is going to be equal to 0. So the overall result is just a combination between 9 and 0. When we add these, we're going to get 9 + 0, which is equal to 9. And that's our final answer. If we consider the answer choices, this corresponds to the answer choice C9. Thank you for watching.

Determine the following limits.
lim x→∞ (5 + (cos⁴ x) / (x² + x + 1))

Key Concepts

Limits at Infinity

Behavior of Trigonometric Functions

Polynomial Growth

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