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

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) = x²(4x² − √(16x⁴ + 1))

Accepted Answer

Hello 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. Identify any vertical asymptotes of the function F of X equals x2 multiplied by 9 multiplied by X2 minus the square root of 81 multiplied by X to the power of 4 + 1. Evaluate the limit as X approaches A from the left of F of X and the limit as X approaches A from the right of F of X for each vertical asymptote X equals A. Awesome. So it appears for this particular problem, we're trying to determine if there are there are any vertical asymptotes for this specific function, and we are also evaluating limits as X approaches A from the left and the limit as X approaches A from the right of F of X for each vertical asymptote where X equals A. Awesome. So now that we know that we're ultimately trying to figure out the vertical asymptotes, let's read off our multiple choice sensors to see what our final answer pair might be. So A is, and I'm going to say vertical asymptote for A through C, but then D, we need to note that D is no vertical asymptote. So A is X equals -2. And the limit as x approaches -2 from the left of F of X is equal to negative infinity, and the limit as X approaches -2 from the right of F of X is equal to infinity. B is X equals 9, and the limit as X approaches 9 from the left of F of X is equal to the limit as X approaches 9 from the right of F of X is equal to negative infinity. And C is X equals 9, and limit as X approaches 9 from the left of F of X is equal to the limit as X approaches 9 from the right of F of X is equal to infinity, and then once again, D is no vertical ascenttotes. OK. So, first off, In order to solve for this particular problem, we need to note and recall right off the right off the bat, we need to note that the given function is defined for all values of X. So, In other words, so it's defined. It's defined for all values. Of X. So because the given function is defined for all values of X, that means that there will be no vertical ascent, so no. Vertical Asymptotes. So that's it, that's it. This problem was very easy to solve because as we should note, we know that our given function is defined for all variables of X. And because it's defined for all variables or all values of X, therefore, as a result, it will not have any vertical asymptotes. So that means without a doubt, our final answer has to be multiple choice sensor letter D, which once again is no vertical asymptotes. 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) = x²(4x² − √(16x⁴ + 1))

Key Concepts

Vertical Asymptotes

Limits

Function Behavior

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