What are the two asymptotes of the hyperbola 𝑦 is equal to five 𝑥 plus one over
three 𝑥 minus four?
In order to find a vertical asymptote here, we need to find the values of 𝑏 such
that any limit as 𝑥 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 𝑥
such that the denominator of 𝑦 is equal to zero. What this mean is that three 𝑥 minus four is equal to zero. Rearranging this, we find that there is a vertical asymptote at 𝑥 is equal to
four-thirds. In order to find the horizontal asymptotes of 𝑦, we need to consider the limit as 𝑥
goes to positive or negative infinity of 𝑦. In order to find the limit as 𝑥 approaches infinity of five 𝑥 plus one over three
𝑥 minus four, we first multiply the numerator and denominator of the fraction by
one over 𝑥.
We are left with the limit as 𝑥 approaches infinity of five plus one over 𝑥 over
three minus four over 𝑥. Then we can use the fact that the limit as 𝑥 approaches infinity of one over 𝑥 is
equal to zero which tells us that one over 𝑥 and negative four over 𝑥 will both
tend to zero as 𝑥 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 𝑥 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 𝑥
equals four-thirds and a horizontal asymptote at 𝑦 equals five-thirds.