# Video: Finding the Asymptotes of a Hyperbola

The graph of equation π¦ = (ππ₯ + π)/(ππ₯ + π) is a hyperbola only if π β  0. In that case, what are the two asymptotes?

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### Video Transcript

The graph of equation π¦ is equal to ππ₯ plus π over ππ₯ plus π is a hyperbola only if π is not on zero. In that case, what are the two asymptotes?

Letβs start by finding the vertical asymptote of this function. Vertical asymptotes occur at π₯ equals π when either the limit as π₯ approaches π from below of π¦ is equal to double negative infinity or the limit as π₯ approaches π from above of π¦ is equal to positive or negative infinity. Since π¦ is a rational function, this will happen at π₯ values, where the denominator of π¦ is equal to zero. Therefore, we can find our vertical asymptotes by setting the denominator of π¦ equal to zero. Now, we solve ππ₯ plus π is equal to zero for π₯. We obtained that thereβs a vertical asymptote as π₯ is equal to negative π over π.

In order to find the horizontal asymptote of π¦, we need to consider the limit as π₯ approaches positive or negative infinity of π¦. In order to find this limit, we can multiply the numerator and denominator by one over π₯. Then we use the fact that the limit as π₯ approaches infinity of one over π₯ is equal to zero, in order to say that when we take this limit, π over π₯ and π over π₯ will both tend to zero. And this leaves us with π over π. Now, we have found our horizontal asymptote. So we have found that the two asymptotes of our hyperbola are π₯ is equal to negative π over π and π¦ is equal to π over π.