Video: Finding the Asymptotes of a Hyperbola

What are the two asymptotes of the hyperbola 𝑦 = (5π‘₯ + 1)/(3π‘₯ βˆ’ 4)?

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Video Transcript

What are the two asymptotes of the hyperbola 𝑦 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 𝑏 such that any limit as π‘₯ approaches 𝑏 of 𝑦 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 𝑦 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 𝑦, we need to consider the limit as π‘₯ goes to positive or negative infinity of 𝑦. 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. Let’s quickly note that if we consider the limit as π‘₯ approaches negative infinity of 𝑦, 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 𝑦 equals five-thirds.

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