End Behavior and Horizontal Asymptotes
Marcus pours a cup of coffee at 180 °F in a kitchen that is 72 °F. Will the coffee ever drop to 0 °F? Obviously not, because it levels off near room temperature. Many functions behave like this when the input gets huge: they settle toward a number, or they blow up. In this page you will learn to predict that long-run behavior without graphing forever.
What happens far from the origin?
End behavior describes what $f(x)$ does when $x$ gets very large in the positive direction ($x \to +\infty$) or very large in the negative direction ($x \to -\infty$). The symbol $\infty$ is not a number. It means "growing without any bound".
Here is the graph of $f(x) = 2 + \dfrac{2}{x}$. Follow it to the right and to the left.
Both ends of the curve hug the dashed line $y = 2$. A table of values shows the same thing:
| $x$ | 10 | 100 | 1,000 | 10,000 |
|---|---|---|---|---|
| $f(x)$ | 2.2 | 2.02 | 2.002 | 2.0002 |
So a function can do one of three things at an end: settle toward a number, grow without bound upward, or fall without bound downward. (A few, like $\sin x$, just keep oscillating and have no limit.)
Look at the following graph of the function $f(x)$. Determine the value the function approaches when $x$ becomes infinitely large ($x \to +\infty$) and when it becomes…
Look at the following graph of the function $f(x)$. Determine the value the function approaches when $x$ becomes infinitely large ($x \to +\infty$) and when it becomes infinitely small ($x \to -\infty$).
The following table shows the values taken by a function $g(x)$ for increasingly larger values of $x$: $x = 10 \rightarrow g(x) = 2,9$ $x = 100 \rightarrow g(x) = 2,99$…
The following table shows the values taken by a function $g(x)$ for increasingly larger values of $x$:
-
$x = 10 \rightarrow g(x) = 2,9$
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$x = 100 \rightarrow g(x) = 2,99$
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$x = 1000 \rightarrow g(x) = 2,999$
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$x = 10000 \rightarrow g(x) = 2,9999$
In light of these results, what value does the function $g(x)$ tend to when $x \to +\infty$?
Polynomials: the leading term wins
Look at $f(x) = 2x^3 - 5x^2 + 7x - 1$ for larger and larger $x$, next to just its first term $2x^3$.
| $x$ | $f(x)$ | $2x^3$ |
|---|---|---|
| 10 | 1,569 | 2,000 |
| 100 | 1,950,699 | 2,000,000 |
| 1,000 | 1,995,006,999 | 2,000,000,000 |
The two columns get closer in relative terms. When $x$ is huge, $x^3$ is far bigger than $x^2$ or $x$, so the term with the highest power drowns out the others. This is why only the leading term matters for end behavior.
You do not need to memorize the table. Just ask: what sign does $a_n x^n$ have when $x$ is huge and positive? When $x$ is huge and negative? An even power is always positive. An odd power keeps the sign of $x$.
Calculate the behavior at infinity of the following polynomial function, that is, find the value of the limit: $\lim_{x \to +\infty} (2x^3 - 5x^2 + 7x - 1)$
Calculate the behavior at infinity of the following polynomial function, that is, find the value of the limit:
$\lim_{x \to +\infty} (2x^3 - 5x^2 + 7x - 1)$
Determine the trend of the function $h(x) = -4x^5 + 3x^2 - 8$ when $x$ tends to $-\infty$. To do this, calculate $\lim_{x \to -\infty} h(x)$.
Determine the trend of the function $h(x) = -4x^5 + 3x^2 - 8$ when $x$ tends to $-\infty$. To do this, calculate $\lim_{x \to -\infty} h(x)$.
Study the behavior of the polynomial function $f(x) = (2x^2 - 3x + 1) - (x^3 - 5x)$ by calculating its limits when $x \to +\infty$ and when $x \to -\infty$.
Study the behavior of the polynomial function $f(x) = (2x^2 - 3x + 1) - (x^3 - 5x)$ by calculating its limits when $x \to +\infty$ and when $x \to -\infty$.
Rational functions: compare the degrees
A rational function is a polynomial divided by a polynomial. Now two leading terms compete: the one on top and the one on the bottom. What matters is which is bigger, or whether they are the same size.
Why? Divide every term of the top and bottom by the highest power of $x$ that appears in the denominator. Every leftover piece like $\frac{5}{x^2}$ or $\frac{1}{x}$ shrinks to 0 as $x$ grows. Only the leading coefficients survive.
Calculate the following limit of a rational function where the degree of the denominator is greater than that of the numerator:…
Calculate the following limit of a rational function where the degree of the denominator is greater than that of the numerator:
$\lim_{x \to +\infty} \frac{3x - 5}{x^2 + 4x + 1}$
Calculate the limit when $x \to -\infty$ of the following rational function, paying attention to the signs and degrees of the polynomials:…
Calculate the limit when $x \to -\infty$ of the following rational function, paying attention to the signs and degrees of the polynomials:
$\lim_{x \to -\infty} \frac{5x^3 - 2x^2 + x}{4 - 3x^3}$
The average production cost of $x$ units of an electronic item is given by the rational function $C(x) = \frac{2500 + 15x}{x}$ in dollars. Calculate the limit of this…
The average production cost of $x$ units of an electronic item is given by the rational function $C(x) = \frac{2500 + 15x}{x}$ in dollars. Calculate the limit of this function when $x \to +\infty$ and interpret the result in the context of the company.
Exponential functions
Exponential functions $y = a \cdot b^x$ are the fastest-moving functions you will meet, and their end behavior depends on the base $b$.
| $x$ | 10 | 20 | 30 |
|---|---|---|---|
| $2^x$ | 1,024 | 1,048,576 | 1,073,741,824 |
| $0.5^x$ | about 0.001 | about 0.000001 | about 0.000000001 |
A base bigger than 1 means repeated multiplication by something larger than 1, so the values explode. A base between 0 and 1 means repeated multiplication by a fraction, so the values shrink toward 0 (but stay positive).
The number $e \approx 2.718$ is just a base bigger than 1. So $e^{-0.02t} = (e^{-0.02})^t$ is a decay factor and goes to 0 as $t \to +\infty$. In general $e^{-kt} \to 0$ when $k > 0$.
Back to the start: the coffee never goes to 0 °F. It levels off at room temperature, 72 °F, which is the horizontal asymptote of its cooling model.
A colony of bacteria grows according to the function $P(t) = 150 \cdot 2^t$, where $t$ is measured in hours. What happens to the bacteria population when time advances…
A colony of bacteria grows according to the function $P(t) = 150 \cdot 2^t$, where $t$ is measured in hours. What happens to the bacteria population when time advances indefinitely ($t \to +\infty$)?
The amount of a radioactive substance in milligrams follows the function $M(t) = 500 \cdot 0,8^t$, where $t$ is the years elapsed. What value does the amount of…
The amount of a radioactive substance in milligrams follows the function $M(t) = 500 \cdot 0,8^t$, where $t$ is the years elapsed. What value does the amount of substance tend to in the very long term ($t \to +\infty$)?
The temperature in degrees Fahrenheit of a freshly poured cup of coffee in a room decreases with time $t$ (in minutes) according to the exponential function…
The temperature in degrees Fahrenheit of a freshly poured cup of coffee in a room decreases with time $t$ (in minutes) according to the exponential function $T(t) = 22 + 68 \cdot 0,85^t$. What temperature does the coffee tend to when a long time has elapsed? Justify your answer by mathematically calculating the corresponding limit.
Horizontal asymptotes in context
In a word problem, the horizontal asymptote answers "what happens in the long run?" Take a food truck with fixed costs of $\$800$ per month and $\$4$ of ingredients per item. The average cost per item for $x$ items is
$$C(x) = \frac{800 + 4x}{x} = 4 + \frac{800}{x}.$$Dividing out, $\frac{800}{x} \to 0$ as $x \to +\infty$, so $\lim_{x \to +\infty} C(x) = 4$. Meaning: the more items the truck makes, the closer the average cost gets to $\$4$ per item (the fixed cost gets spread thinner and thinner). The cost will never fall below $\$4$.
When you interpret an asymptote, always say: the quantity, which value it approaches, what "$x$ large" means, and (if relevant) whether it can be exceeded.
Finding unknown parameters
If a problem tells you the asymptote, use the limit rule backwards.
Look at the graph of the function $C(x)$ representing the average cost (in dollars) of manufacturing $x$ objects. State the equation of the horizontal asymptote and…
Look at the graph of the function $C(x)$ representing the average cost (in dollars) of manufacturing $x$ objects. State the equation of the horizontal asymptote and explain its meaning when the number of objects produced tends to infinity.
Find the value of the parameter $k$ knowing that the rational function $f(x) = \frac{kx^2 - 3x + 7}{4x^2 + 5}$ has a horizontal asymptote at the line $y = 3$.
Find the value of the parameter $k$ knowing that the rational function $f(x) = \frac{kx^2 - 3x + 7}{4x^2 + 5}$ has a horizontal asymptote at the line $y = 3$.
Determine the values of the parameters $a$ and $b$ of the rational function $f(x) = \frac{ax + 8}{bx - 2}$ knowing that its graph passes through the point $(1, -10)$ and…
Determine the values of the parameters $a$ and $b$ of the rational function $f(x) = \frac{ax + 8}{bx - 2}$ knowing that its graph passes through the point $(1, -10)$ and has a horizontal asymptote at the line $y = 3$.
Challenges: absolute values, "how close", and growth races
Absolute value and piecewise functions
Remember $|x| = x$ when $x > 0$ and $|x| = -x$ when $x < 0$. So for $x \to +\infty$ replace $|x|$ by $x$, and for $x \to -\infty$ replace $|x|$ by $-x$. Piecewise functions work the same way: use the piece that applies for large positive $x$, or the piece that applies for large negative $x$.
Exponential functions can also give two asymptotes. For $h(x) = 3 - \dfrac{6}{1 + 2^x}$: as $x \to +\infty$, $2^x \to \infty$, so $\frac{6}{1 + 2^x} \to 0$ and $h \to 3$. As $x \to -\infty$, $2^x \to 0$, so $h \to 3 - 6 = -3$. The asymptotes are $y = 3$ and $y = -3$.
Parameters with a point
Sometimes you must combine the asymptote with a point on the graph. Example: $f(x) = \dfrac{ax + 4}{bx - 1}$ has asymptote $y = 2$ and passes through $(1, 6)$. The asymptote gives $\frac{a}{b} = 2$, so $a = 2b$. The point gives $\frac{a + 4}{b - 1} = 6$. Substitute $a = 2b$: $2b + 4 = 6(b - 1)$, so $10 = 4b$, giving $b = 2.5$ and $a = 5$. Check: $f(1) = \frac{9}{1.5} = 6$. ✓
How close is close?
"$f(x) \to L$" means you can make $|f(x) - L|$ as small as you want by taking $x$ large enough. Here is how to find how large.
If the inequality turns out to be quadratic (like $\frac{30t}{t^2 + 9} < 1$ in the drug problem), rearrange it to $t^2 - 30t + 9 > 0$, find the roots with the quadratic formula, and pick the region that makes sense for the context (here, after the concentration peaks).
Which grows faster?
Compare $x^3$ and $2^x$. At $x = 5$ the polynomial is bigger ($125$ vs $32$), but watch the ratio $\frac{x^3}{2^x}$:
| $x$ | 5 | 10 | 20 | 30 |
|---|---|---|---|---|
| $x^3$ | 125 | 1,000 | 8,000 | 27,000 |
| $2^x$ | 32 | 1,024 | 1,048,576 | 1,073,741,824 |
| $\dfrac{x^3}{2^x}$ | 3.9 | 0.98 | 0.0076 | 0.000025 |
The exponential wins and the ratio shrinks toward 0. So $\lim_{x \to +\infty} \frac{x^3}{2^x} = 0$.
Let the rational function be $f(x) = \frac{3x}{x+2}$. Calculate its limit $L$ as $x \to +\infty$. Next, set up and solve an inequality to find for what values of $x > 0$…
Let the rational function be $f(x) = \frac{3x}{x+2}$. Calculate its limit $L$ as $x \to +\infty$. Next, set up and solve an inequality to find for what values of $x > 0$ it is true that the distance between $f(x)$ and its limit $L$ is strictly less than $0,01$.
Study the behavior at infinity of the function containing an absolute value $f(x) = \frac{4x + |x|}{x + 2}$. To do this, calculate the limits as $x \to +\infty$ and as…
Study the behavior at infinity of the function containing an absolute value $f(x) = \frac{4x + |x|}{x + 2}$. To do this, calculate the limits as $x \to +\infty$ and as $x \to -\infty$. Does the function have the same horizontal asymptote on the right as on the left?
Consider the functions $f(x) = x^4$ and $g(x) = 1,5^x$. For small values of $x$, the polynomial function is greater than the exponential one (for example, for $x=2$,…
Consider the functions $f(x) = x^4$ and $g(x) = 1,5^x$. For small values of $x$, the polynomial function is greater than the exponential one (for example, for $x=2$, $16 > 2,25$). However, determine which of the two grows faster at infinity by using a table of values for $x = 10, 20$ and $30$. From your conclusion, deduce the value of the limit $\lim_{x \to +\infty} \frac{x^4}{1,5^x}$.