Find the vertical asymptotes. For each vertical asymptote x=a,...

Question

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=cos x+2√x / √x.

Accepted Answer

Hello there. Today we're gonna 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 vertical asymptotes of the function J X equals 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 asked to solve for the vertical asymptotes of this specific function. So now that we know that we're ultimately trying to figure out the vertical asymptotes for this specific function, let us read off our multiple choice answers to see what our final answer might be, noting that it states that X is equal to some value. A is 0, B is 1 and -1, C is -1, and D is 1. OK. So first off, in order to help us solve for this problem, we need to set the denominator equal to 0 and solve for X. So when we take the square root of X is equal to 0, we will determine that X is equal to 0. And now our next step is we need to check, we need to check this equation to see if the left hand and right hand limit of the function when X approaches 0 are finite or infinite. So let me say that again. So now our next step is we need to check if the left hand limit. And the right hand limit of the function when X approaches 0, so let's make a note of that. So this is when X approaches 0, are finite or infinite. So with that in mind, let us take the limit. We need to take the limit as X approaches 0 from the right, and that's what the little plus sign means and the power. This plus sign means it's from the right. So the limit as X approaches 0 from the right of Cosine of X plus 5 multiplied by the square root of X, divided by the square root of X will be equal to infinity. So, since, as a result, since one of the limits when X approaches 0 is infinite, That will mean as a result, we will have a vertical asymptote at X equals 0, and that's it. We've solved for this problem. Hooray. So to quickly recap here, So as a result, since we had one of the limits, as we should recall, when X approaches 0 is equal to infinity, that will mean that the vertical ascentote will be X equals 0, and this corresponds with multiple choice answer letter A, which once again is X equals 0. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=cos x+2√x / √x.

Key Concepts

Vertical Asymptotes

Limits

Rational Functions

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