Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any ...

Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=cos x+2√x / √x.

Accepted Answer

Below there today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the horizontal asymptotes of the function J of X is equal to cosine of X plus 5 multiplied by the square root of X, all divided by the square root of X. Awesome. So it appears for this particular problem we're trying to figure out the horizontal asymptotes for this specific function. So now that we know that we're ultimately trying to figure out the horizontal asymptotes for this function that's given to us, let's read off our multiple choice sensors to see what our final answer might be, noting that they all state that Y equals some number. So A equals -5 and 5. B is -5, C is 5, and D is 10. OK. So first off, we need to recall and write that the limit as X approaches negative infinity. We can write that the limit as X approaches negative infinity of cosine of X divided, so it's the cosine of X, and we need to add plus. 5 multiplied by the square root of X, all divided by the square root of X is equal to the limit as X approaches negative infinity of cosine of X divided by the square root of X. Plus 5 multiplied by the square root of X, divided by the square root of X. And this is all equal to the limit, as X approaches negative infinity of cosine of X divided by the square root of X plus the limit. And when you take the limit as X approaches negative infinity. Of 5, and this will, as we should recall, will be equal to D and E where DNE means does not exist. So this big old long expression, when we were evaluating the limit as X approaches negative infinity, it will yield us a does not exist value. So this means that the function is not defined for X is less than 0. So let's write that in red. So that means that it is not defined, so I've to ND not defined where X is less than 0, because it involves a square root of X, and because it involves a square root of X, it will not be real for a negative value of X. So this is not real for a negative value of X. So, moving forward or moving on here, we need to also recall that we should be able to write that the limit as X approaches infinity. So the limit as X approaches infinity of cosine of X plus. 5 multiplied by the square root of X divided by the square root of X. is equal to the limit as X approaches, and now we're dealing with positive infinity, we're not dealing with negative infinity, of cosine of X divided by the square root of X. Plus, 5 multiplied by the square root of X divided by the square root of X, and this will be all equal to the limit as X approaches infinity of cosine of X divided by the square root of X. Plus the limit as X approaches infinity of 5. And this will be all equal to 0 + 5, which will mean that our final answer has to be equal to 5 for this expression. And once again, this is when we evaluate our limit. So when we take the limit of the function as X approaches infinity, and we also took our function and we applied the limit where X approaches negative infinity. So that's what we did just now. So, now at this point, we need to recall and note that for cosine of X divided by the square root of X, we need to note that as X approaches infinity, We need to know that the square root of X will become infinitely large, so I put an arrow pointing upwards since it'll become infinitely larger. So this will make the fraction approach 0. And because it approaches 0, this is because, so this is approaching 0 due to the fact that the cosine of X will be oscillating between -1 and 1. So therefore, as a result, its magnitude will be bounded. So therefore, without a doubt, our horizontal asymptote will be equal to, so Y will be equal to 5, and that is our horizontal asymptote. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be the letter C, which once again is Y is equal to 5. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=cos x+2√x / √x.

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Rational Functions and Simplification

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