What does a limit at infinity tell me about a function’s long-run behavior?
Quick answer
A finite limit at infinity tells you which value a function’s outputs approach as its input grows without bound. If the limit is L, the graph has horizontal asymptote y = L. The function does not have to equal L or stop at that line. For example, 1/x approaches 0 as x approaches positive infinity, so y = 0 is its horizontal asymptote. [S1]
The question, the details, the next step.
When I see a limit as x goes to infinity, what am I meant to learn about the function?
You’re learning about the function’s long-run behavior: as x grows larger and larger, what number do the function values get closer to? If those values approach a finite number L, we write the limit as L. The graph then has horizontal asymptote y = L. [S1]
This describes what happens far to the right on the graph. It does not say that the function must take the value L. The asymptote describes the trend in the function’s values as the input grows, not a point the graph has to land on. [S1]
Can you show me how to find the value with a simple rational function?
Consider f(x) = 1/x. As x approaches positive infinity, the denominator grows while the numerator stays 1, so the fraction approaches 0. In limit notation:
lim (x→+∞) 1/x = 0.
The limiting value is 0, so the graph has horizontal asymptote y = 0. This is a rational-function example of a finite limit at infinity. [S1]
The notation x→+∞ means we are considering increasingly large positive inputs. The limit describes the values the function approaches in that direction; it is not a statement about what happens for every input.
Does this function actually equal 0, or is it only approaching 0?
For f(x) = 1/x, it does not equal 0 at any input where the function is defined: a fraction with numerator 1 cannot have value 0. Its values approach 0 as x grows, without reaching it.
That is true for this example, but it is not a rule for every function. A function can equal its limiting value or even cross its horizontal asymptote. The limit tells you about long-run behavior; it does not, by itself, decide whether the function reaches that value elsewhere. [S1]
What should I write to explain the answer, and what should I keep in mind?
State the limit, name the horizontal asymptote, and explain the meaning of the limit. For this example, you could write:
“As x approaches positive infinity, 1/x approaches 0. Therefore, y = 0 is the horizontal asymptote. The function’s values get closer to 0 as x grows, but the function does not equal 0.”
The last part is specific to 1/x; don’t assume every function behaves that way. A limit at infinity describes the trend in one direction, not every part of the graph. [S1]
A practical checklist
Identify the direction of the limit, such as x approaching positive infinity.
Find the finite value the function approaches, if there is one.
State the horizontal asymptote as y = that value.
Explain the long-run trend in words.
Check separately whether the function reaches or crosses the limiting value; it is not ruled out by the limit alone.
A few more useful answers
Does a limit at infinity describe the function at one particular input?
No. It describes the trend in the function’s values as the input grows without bound, rather than its value at one specific input.
Does a horizontal asymptote mean the graph can never cross that line?
No. A function can cross its horizontal asymptote; the asymptote describes long-run behavior, not a ban on crossing. [S1]
Does a function have to reach its limiting value?
No. A function can approach a finite limiting value without equaling it. Whether it reaches or crosses that value depends on the function. [S1]
Sources
References used for this guide.
- [S1] OpenStax Calculus Volume 1, Section 4.6: Limits at Infinity and Asymptotes openstax.org · Checked