Video Transcript
Find the horizontal asymptotes of the function 𝑓 of 𝑥 is equal to three 𝑥 squared plus seven divided by five 𝑥 squared minus four.
The question wants us to find the horizontal asymptotes of our rational function 𝑓 of 𝑥. And we recall, we say that the line 𝑦 is equal to 𝑙 is a horizontal asymptote to the curve 𝑦 is equal to 𝑓 of 𝑥 if the limit as 𝑥 approaches ∞ of 𝑓 of 𝑥 is equal to 𝑙 or the limit as 𝑥 approaches negative ∞ of 𝑓 of 𝑥 is equal to 𝑙. So to find the possible horizontal asymptotes of our function, we just need to calculate the limit as 𝑥 approaches ∞ and the limit as 𝑥 approaches negative ∞ of our rational function 𝑓 of 𝑥.
Let’s start by calculating the limit as 𝑥 approaches ∞ of 𝑓 of 𝑥. That’s the limit as 𝑥 approaches ∞ of three 𝑥 squared plus seven divided by five 𝑥 squared minus four. If we tried to evaluate this limit directly, we would see that we would get ∞ divided by ∞. This is an indeterminate form. So we’ll have to evaluate this limit in a different way.
We can actually do this by dividing both our numerator and our denominator by the highest power of 𝑥 which appears in the fraction. We see that the highest power of 𝑥 which appears in our fraction is 𝑥 squared. So we’re going to divide both our numerator and our denominator through by 𝑥 squared.
In our numerator, we have three 𝑥 squared divided by 𝑥 squared is just three. And seven divided by 𝑥 squared is seven over 𝑥 squared. In our denominator, we have five 𝑥 squared divided by 𝑥 squared is five. And then negative four divided by 𝑥 squared is negative four divided by 𝑥 squared. And we can now see by evaluating this limit since 𝑥 is approaching ∞, seven divided by 𝑥 squared and negative four divided by 𝑥 squared are both approaching zero.
And we know that our constants of three and five are not changing with 𝑥. So we can actually just evaluate this limit to give us three divided by five. We could then write out and evaluate the limit as 𝑥 approaches negative ∞ of 𝑓 of 𝑥 in a similar way. However, we could also ask the question, what would’ve happened if, instead of we’d taken the limit as 𝑥 approaches ∞, we’d taken the limit as 𝑥 approaches negative ∞?
If we’d instead taken the limit as 𝑥 approaches negative ∞, we can still rewrite 𝑓 of 𝑥 as three 𝑥 squared plus seven divided by five 𝑥 squared minus four. And we can still divide through our numerator and our denominator by 𝑥 squared. And dividing through by 𝑥 squared does not change.
We now need to calculate the limit as 𝑥 approaches negative ∞ of three divided by seven over 𝑥 squared all divided by five minus four over 𝑥 squared. And we can see the same line of reasoning holds. Seven over 𝑥 squared and negative four over 𝑥 squared still both approach zero as 𝑥 approaches negative ∞. And three and five are still both constants even when 𝑥 is approaching negative ∞. So the limit as 𝑥 approaches negative ∞ of 𝑓 of 𝑥 is also equal to three divided by five.
Therefore, we’ve shown that the only horizontal asymptote of our function 𝑓 of 𝑥 is equal to three 𝑥 squared plus seven all divided by five 𝑥 squared minus four is the line 𝑦 is equal to three-fifths.
