Module 3 · Lesson 3.3

Limits at Infinity & End Behavior

See how a curve flattens toward a horizontal asymptote as $x$ grows without bound, and read the limiting value that describes that end behavior.

Intuition

Walk far enough along the positive $x$-axis and many curves stop climbing or falling; they level off. The height they settle toward is a horizontal asymptote. The mathematical statement “the limit as $x$ approaches infinity equals $L$” simply records that eventual flattening.

The same picture appears in physics: a skydiver accelerates at first, then air resistance balances gravity and velocity approaches a constant terminal value. The velocity graph flattens exactly the way a rational function approaches its horizontal asymptote.

Core concept

A limit at infinity describes end behavior. If the $y$-values of a function get arbitrarily close to a number $L$ once $x$ is large enough (positive or negative), we write $\lim_{x\to\infty}f(x)=L$ or $\lim_{x\to-\infty}f(x)=L$. Graphically the curve is flattening toward the horizontal line $y=L$.

Function$f(x)=\dfrac{2x}{x+1}$
At $x=1$$f=1$
At $x=10$$f\approx 1.818$
At $x=100$$f\approx 1.980$
As $x\to\infty$$f\to 2$

Interactive Lab

Slide $x$ farther to the right. Watch the curve approach the green horizontal asymptote and read how the function value settles toward 2.

$f(x)$: 1.600 Gap to 2: 0.400
x y y = 2

Function $f(x)=\frac{2x}{x+1}$. As $x$ grows, the $+1$ in the denominator becomes negligible, so $f(x)$ behaves like $\frac{2x}{x}=2$. The green dashed line is the horizontal asymptote that the curve never quite reaches.

Key Idea

$$ \lim_{x \to \infty} f(x) = L $$

The statement means that for any tolerance $\varepsilon>0$ there exists a large enough $M$ such that once $x>M$, the distance $|f(x)-L|$ is smaller than $\varepsilon$. In practice we often divide numerator and denominator by the highest power of $x$ that appears; the dominant terms reveal the constant $L$ that the function approaches. The same idea works for $x\to-\infty$.

$x\to\infty$$x$ grows without bound in the positive direction
$L$The limiting height the curve approaches (horizontal asymptote $y=L$)
$f(x)$The function whose end behavior we are reading
$|f(x)-L|$The vertical gap that shrinks as $x$ becomes large

Think Further

  1. Why can a curve approach a horizontal asymptote yet never actually touch it?
  2. In the terminal-velocity example, what physical quantity plays the role of the horizontal asymptote $L$?
  3. If the degrees of the numerator and denominator of a rational function are equal, what is the horizontal asymptote in terms of the leading coefficients?
Show suggested answers
  1. An asymptote describes what happens as $x$ grows without bound; it does not require equality at any finite input.
  2. $L$ is the terminal speed approached when drag balances the driving force.
  3. The asymptote is the ratio of the leading coefficients, $y=\frac{a}{b}$.