Visually determining vertical asymptotes | Limits | Differential Calculus | Khan Academy

2m read

·Nov 11, 2024

Given the graph of yal ( f(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 = -4 ) and ( x = 2 ). At ( x = -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 = -4 ) from the left, the value of our function goes to infinity. Likewise, as we approach ( x = -4 ) from the right, it looks like our value of our function goes to infinity.

So I'd say that we definitely have a vertical asymptote at ( x = -4 ). Now let's look at ( x = 2 ). As we approach ( x = 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 = 2 ) from the right, it looks like we’re approaching ( f(x) = -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 = 2 ) a vertical asymptote.

So we can say that there's a vertical asymptote at ( x = -4 ) and ( x = 2 ).

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