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Studying for a test? Prepare with these 21 lessons on Limits and continuity.
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Given the graph of y equals f of x pictured below, determine the equations of all vertical asymptotes. So let's see what's going on here. So it looks like interesting things are happening at x equals negative 4 and x equals 2. At x equals negative 4, as we approach it from the left, the value of the function just becomes unbounded right over here. It looks like as we approach x equals negative 4 from the left, the value of our function goes to infinity. Likewise, as we approach x equals negative 4 from the right, it looks like the value of our function goes to infinity. So I'd say that we definitely have a vertical asymptote at x equals negative 4. Now let's look at x equals 2. As we approach x equals 2 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 2 from the right, it looks like we're approaching f of x is equal to negative 4. 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 2 a vertical asymptote. So we could say that there's a vertical asymptote at x equals negative 4 and x equals 2.