Video: Multiplying Complex Numbers

Video Transcript

Find the horizontal and vertical asymptotes of the function 𝑓 of 𝑥 equals three 𝑥 minus sin 𝑥.

We recall the horizontal line 𝑦 equals 𝐿 is an asymptote to the function 𝑦 equals 𝑓 of 𝑥 if the limit as 𝑥 approaches either positive or negative ∞ of 𝑓 of 𝑥 is equal to 𝐿. Then, for a vertical asymptote, we say 𝑥 equals 𝑎 is an asymptote if the limit as 𝑥 approaches 𝑎 from the left of 𝑓 of 𝑥 is either positive or negative ∞. If the limit as 𝑥 approaches 𝑎 from the right of 𝑓 of 𝑥 is either positive or negative ∞. Or if the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is either positive or negative ∞.

Now, in this question, 𝑓 of 𝑥 is given by three 𝑥 minus sin 𝑥. So let’s begin by finding the limit as 𝑥 approaches ∞ of three 𝑥 minus sin 𝑥. Then, we recall that the limit of the sum or difference of two functions is equal to the sum or difference of the limits of those respective functions. And we can write this as the limit as 𝑥 approaches ∞ of three 𝑥 minus the limit as 𝑥 approaches ∞ of sin 𝑥. As 𝑥 approaches ∞, three 𝑥 itself also approaches ∞ whereas sin 𝑥 can only take values in the closed interval from negative one to one. And so the limit as 𝑥 approaches ∞ of three 𝑥 minus sin 𝑥 is ∞. This isn’t a constant value as required of 𝐿. So there’s no horizontal asymptote here.

We should check the limit as 𝑥 approaches negative ∞. Once again, we split it up into the limit of three 𝑥 minus the limit of sin 𝑥. As 𝑥 approaches negative ∞, three 𝑥 itself also approaches negative ∞. Once again though, sin 𝑥 oscillates between negative one and one. This has very little impact on a number as large as negative ∞. So the limit as 𝑥 approaches negative ∞ of three 𝑥 minus sin 𝑥 is negative ∞. And we can say that there are no horizontal asymptotes.

Now, we’ll consider the vertical asymptotes. We’ll split our limit up. And we’ll look for the limit as 𝑥 approaches 𝑎 of three 𝑥 minus the limit as 𝑥 approaches 𝑎 of sin 𝑥. We want to find a situation where this might be equal to either positive or negative ∞. Well, we saw that the only way for the limit of three 𝑥 to be ∞ is if 𝑥 itself approaches ∞. Similarly, for three 𝑥 to approach negative ∞, 𝑥 itself must approach negative ∞ whereas the range of sin 𝑥 is the closed interval from negative one to one. So the only way for the limit of our function to approach positive or negative ∞ is if 𝑥 itself approaches positive or negative ∞. So actually, there are no vertical asymptotes either. And so the function 𝑓 of 𝑥 equals three 𝑥 minus sin 𝑥 has no horizontal or vertical asymptotes.