Video: Finding the Asymptotes of a Hyperbola

What are the two asymptotes of the hyperbola π¦ = (5π₯ + 1)/(3π₯ β 4)?

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Video Transcript

What are the two asymptotes of the hyperbola π¦ is equal to five π₯ plus one over three π₯ minus four?

In order to find a vertical asymptote here, we need to find the values of π such that any limit as π₯ approaches π of π¦ is equal to positive or negative infinity. In order to find the vertical asymptotes, we simply need to find the values of π₯ such that the denominator of π¦ is equal to zero. What this mean is that three π₯ minus four is equal to zero. Rearranging this, we find that there is a vertical asymptote at π₯ is equal to four-thirds. In order to find the horizontal asymptotes of π¦, we need to consider the limit as π₯ goes to positive or negative infinity of π¦. In order to find the limit as π₯ approaches infinity of five π₯ plus one over three π₯ minus four, we first multiply the numerator and denominator of the fraction by one over π₯.

We are left with the limit as π₯ approaches infinity of five plus one over π₯ over three minus four over π₯. Then we can use the fact that the limit as π₯ approaches infinity of one over π₯ is equal to zero which tells us that one over π₯ and negative four over π₯ will both tend to zero as π₯ tends to infinity. And so, therefore, we find that our limit is equal to five-thirds. Letβs quickly note that if we consider the limit as π₯ approaches negative infinity of π¦, then we would see that this limit is also equal to five-thirds. Therefore, the solution to this question is that we have a vertical asymptote at π₯ equals four-thirds and a horizontal asymptote at π¦ equals five-thirds.