Calculus, all content (2017 edition)
- Infinite limits intro
- Connecting limits at infinity notation and graph
- Infinite limits: graphical
- Analyzing unbounded limits: rational function
- Analyzing unbounded limits: mixed function
- Infinite limits: algebraic
- Visually determining vertical asymptotes (old)
An old video where Sal identifies all the vertical asymptotes of a function given graphically. Created by Sal Khan.
- [Instructor] Given the graph of y equals f of x pictured below, determine the equations of all vertical asymptotes. Let's see what's going on here. So it looks like interesting things are happening at x equals negative four and x equals two. At x equals negative four, as we approach it from the left the value of the function just becomes unbounded right over here. Looks like as we approach x equals negative four from the left, the value of our function goes to infinity. Likewise, as we approach x equals negative four from the right, it looks like our, the value of our function goes to infinity. So, I'd say that we definitely have a vertical asymptote at x equals negative four. Now, let's look at x equals two. As we approach x equals two from the left, the value of our function once again approaches infinity, or it becomes unbounded. Now, from the right, we have an interesting thing. If we look at the limit from the right right over here, it looks like we're approaching a finite value. As we approach x equals two from the right, it looks like we're approaching f of x is equal to negative four. But, just having a one-sided limit that is unbounded, is enough to think about this as a vertical asymptote. The function is not defined right over here, and as we approach it from just one side, we are becoming unbounded. It looks like we're approaching infinity or negative infinity, so that by itself, this unbounded left hand limit, or left-side limit by itself is enough to consider x equals two a vertical asymptote. So, we can say that there's a vertical asymptote at x equals negative four and x equals two.