AP® Calculus AB · Unit 1 · Topic 1.14
Connecting Infinite Limits and Vertical Asymptotes
A vertical asymptote happens when a function's denominator shrinks to zero while the numerator stays nonzero, forcing the output to blast off toward positive or negative infinity. Saying the limit "is infinity" describes that runaway behavior, not an actual number the function reaches.
A function shoots up or down near some x-value when you're dividing by something that's vanishing. Picture 1/(x-3). As x creeps toward 3, the denominator x-3 shrinks toward 0, and dividing a fixed nonzero top by a tinier and tinier bottom makes the whole fraction explode in size. That explosion is what a vertical asymptote looks like on a graph: the curve hugs the vertical line x=3 and races off the top or bottom of the page without ever touching it.
The direction of the explosion depends on sign. Just left of x=3, x-3 is a small negative number, so 1/(x-3) is a huge negative number — the limit from the left is negative infinity. Just right of x=3, x-3 is small and positive, so 1/(x-3) is huge and positive — the limit from the right is positive infinity. Because these one-sided limits disagree, the two-sided limit at x=3 does not exist, even though each side has a clear infinite trend. That's the key catch: writing 'limit equals infinity' is shorthand for unbounded growth, not a claim that the function settles on some real number called infinity. A limit existing means approaching a specific finite value; infinite limits are a special case of not existing, described in more detail than the plain 'does not exist' verdict.
Not every zero in a denominator makes an asymptote — only the ones that survive after simplifying. If a factor cancels with the numerator, you get a hole, not a wall. Infinite limits at a finite x-value give vertical asymptotes; limits at infinity (letting x itself grow without bound) give horizontal asymptotes instead, describing the function's long-run flat behavior rather than its blow-up behavior.
Key terms
- Vertical asymptote
- A vertical line x=a that a graph approaches but never crosses, where the function's value grows without bound.
- Infinite limit
- A limit description for when a function's output grows arbitrarily large (positive or negative) as x approaches some value.
- Removable discontinuity (hole)
- A point missing from a graph because a factor canceled top and bottom, rather than causing unbounded growth.
- One-sided limit
- The trend of a function's values as x approaches a point from only the left or only the right.
Watch it explained
- How it worksHow does a denominator heading to zero while the top stays put create a vertical wall?
- Worked exampleFind the one-sided limits of 1/(x - 3) as x approaches 3 from each side.
- Common mistakeWhy isn't a limit of infinity the same thing as the limit existing?
- Ties togetherHow does a vertical asymptote from an infinite limit compare with a horizontal asymptote from a limit at infinity?
Check yourself
End of topic.