End Behavior and Horizontal Asymptotes
Emma pulls a pizza out of the oven at 200 °F. The kitchen is 70 °F, and the pizza cools according to $T(t) = 70 + 130 \cdot 0.9^t$, where $t$ is in minutes. If she waits "forever," will the pizza cool down to 0 °F? Or does it stop somewhere? To answer, we need to know what a function does when $x$ (or $t$) gets enormous. That is called end behavior.
What happens "at infinity"?
The symbol $x \to +\infty$ means "$x$ keeps getting larger without stopping": 1,000, then 1,000,000, then 1,000,000,000, and so on. Likewise, $x \to -\infty$ means $x$ keeps getting more negative. Infinity is not a number you can reach. It describes a direction.
We ask: as $x$ goes in that direction, what does $f(x)$ do? There are three possible answers.
Here is a graph that settles down near $y = 3$ on the right.
You can also detect a limit from a table. Take $h(x) = 5 + \dfrac{7}{x}$:
| $x$ | 10 | 100 | 1,000 | 10,000 |
|---|---|---|---|---|
| $h(x)$ | 5.7 | 5.07 | 5.007 | 5.0007 |
The values get closer and closer to 5, so $\lim_{x \to +\infty} h(x) = 5$ and $y = 5$ is a horizontal asymptote.
In applications, a horizontal asymptote is a long-run value. If $C(x)$ is the average cost in dollars per item after making $x$ items, and the graph levels off at $y = 12$, then the average cost per item gets closer and closer to $\$12$ as production grows without bound.
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 $x$:
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$x = 10 \rightarrow g(x) = 2,9$
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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 $x \to +\infty$?
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.
End behavior of polynomials
Look at $f(x) = 2x^3 - 5x^2 + 7x - 1$ for larger and larger $x$, next to 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 first term takes over. When $x$ is huge, $x^3$ is far bigger than $x^2$ or $x$, so the other terms barely matter. A polynomial behaves like its leading term (the term with the highest power) at the ends.
Slide $a$ and watch the two ends flip when the sign of the leading coefficient changes.
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 fraction of two polynomials. If both top and bottom grow, who wins? The key fact is that for any positive integer $n$, $\dfrac{1}{x^n} \to 0$ as $x \to \pm\infty$. For example, $\frac{1}{1000} = 0.001$ and $\frac{1}{1{,}000{,}000} = 0.000001$.
The method is to divide every term, top and bottom, by the highest power of $x$ in the denominator. Then most pieces turn into $\frac{1}{x^n}$ and vanish.
Doing the same division in general gives a shortcut.
Both degrees are 1, so the asymptote is $y = \frac{a}{1} = a$. Move the slider and check.
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.
Using asymptotes: long-run values and unknown parameters
The limit at infinity answers "what happens in the long run?" Always finish with a sentence that says what the number means in the situation.
Sometimes the asymptote is given, and you must find a missing number. Since the asymptote depends only on the leading coefficients, set that ratio equal to the given value. If the graph also passes through a point, substitute the point into the function.
In an isolated ecosystem, the population of a certain bird species is modeled by the function $P(t) = \frac{1200t + 400}{2t + 5}$, where $t$ is the time in years since…
In an isolated ecosystem, the population of a certain bird species is modeled by the function $P(t) = \frac{1200t + 400}{2t + 5}$, where $t$ is the time in years since records began. Determine what value the bird population will stabilize at in the long term.
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 $y = 3$.
Exponential functions at infinity
For $y = a \cdot b^t$ (with $a > 0$), what happens depends on the base $b$.
| Base | Type | As $t \to +\infty$ | As $t \to -\infty$ |
|---|---|---|---|
| $b > 1$ | growth | $b^t \to +\infty$ | $b^t \to 0$ |
| $0 < b < 1$ | decay | $b^t \to 0$ | $b^t \to +\infty$ |
Why: multiplying by a number bigger than 1 again and again makes things bigger ($2^{10} = 1024$). Multiplying by a number between 0 and 1 again and again makes things smaller ($0.8^{10} \approx 0.107$ and $0.8^{50} \approx 0.0000143$). On the left, $2^{-10} = \frac{1}{1024}$ is tiny.
Slide $b$ below 1 and the curve levels off at $c$. Slide it above 1 and it shoots up.
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.
A saline solution begins to be poured into an industrial tank containing $500$ gallons of pure water at a rate of $10$ gallons per minute, with a concentration of $40$…
A saline solution begins to be poured into an industrial tank containing $500$ gallons of pure water at a rate of $10$ gallons per minute, with a concentration of $40$ ounces of salt per gallon. At the same time, the well-mixed solution leaves the tank at the same rate to keep the volume constant. The salt concentration in the tank (in oz/gal) at time $t$ (in minutes) is given by the function $C(t) = 40(1 - e^{-0,02t})$. Calculate the limit of $C(t)$ as $t \to +\infty$ and explain the physical meaning of this result.
Exponential growth: models and races
When $b > 1$ the limit is $+\infty$. That is the math. But is it realistic?
Now a different question: which grows faster, a polynomial or an exponential? Take $f(x) = x^4$ and $g(x) = 1.5^x$. At $x = 2$, $f(2) = 16$ is way bigger than $g(2) = 2.25$. Let's go far out and compare the ratio $\dfrac{x^4}{1.5^x}$:
| $x$ | $x^4$ | $1.5^x$ | $\dfrac{x^4}{1.5^x}$ |
|---|---|---|---|
| 10 | 10,000 | 57.7 | 173 |
| 20 | 160,000 | 3,325 | 48 |
| 30 | 810,000 | 191,751 | 4.2 |
| 40 | 2,560,000 | about 11,057,000 | 0.23 |
| 50 | 6,250,000 | about 637,600,000 | 0.0098 |
Up to $x = 30$ the polynomial is still ahead. But the ratio keeps falling, and past about $x = 35$ the exponential wins and never gives the lead back.
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 ($t \to +\infty$)?
A principal of $\$2000$ is invested at an annual compound interest rate of $4\%$. The function describing the accumulated capital after $t$ years is $C(t) = 2000 \cdot…
A principal of $\$2000$ is invested at an annual compound interest rate of $4\%$. The function describing the accumulated capital after $t$ years is $C(t) = 2000 \cdot 1,04^t$. What is the limit of $C(t)$ as $t \to +\infty$? Do you consider this mathematical model to be realistic for extremely long periods of time? Explain your answer.
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 $\lim_{x \to +\infty} \frac{x^4}{1,5^x}$.
Two different ends: piecewise and absolute value
The left end and the right end of a graph do not have to match. A function can have two different horizontal asymptotes, or an asymptote on one side only.
For a piecewise function, use only the piece that applies near the end you care about. For $x \to +\infty$, use the rule for large positive $x$. For $x \to -\infty$, use the rule for very negative $x$.
For an absolute value, split into cases. For $x > 0$, $|x| = x$. For $x < 0$, $|x| = -x$.
An exponential can also produce two asymptotes. Take $f(x) = \dfrac{4 \cdot 2^x}{2^x + 1}$. Divide top and bottom by $2^x$ to get $\dfrac{4}{1 + 2^{-x}}$. As $x \to +\infty$, $2^{-x} \to 0$, so $f \to 4$. As $x \to -\infty$, $2^x \to 0$ in the original form, so $f \to \frac{0}{0 + 1} = 0$.
Given the following piecewise function: $f(x) = \begin{cases} \frac{4x^2 + 1}{x^2 - 3} & \text{if } x > 0 \\ 3^x + 2 & \text{if } x \leq 0 \end{cases}$ Calculate the…
Given the following piecewise function:
$f(x) = \begin{cases} \frac{4x^2 + 1}{x^2 - 3} & \text{if } x > 0 \\ 3^x + 2 & \text{if } x \leq 0 \end{cases}$
Calculate the limits $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$.
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?
Look at the following graph representing a function $f(x)$ with two different horizontal asymptotes. From the graph, deduce the value of $\lim_{x \to +\infty} f(x)$ and…
Look at the following graph representing a function $f(x)$ with two different horizontal asymptotes. From the graph, deduce the value of $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$. Based on what you have studied about exponential functions, write a possible algebraic expression that exhibits this type of double asymptotic behavior.
How close does it get? Tolerances and caps
Saying "$f(x) \to L$" is a promise: you can make $f(x)$ as close to $L$ as you like by taking $x$ large enough. To make that concrete, pick a tolerance $\varepsilon$ (for example $0.05$) and ask, "For which $x$ is $|f(x) - L| < \varepsilon$?"
The shaded band is the allowed tolerance. Once the curve enters the band, it stays inside.
Sometimes the inequality turns into a quadratic.
A related question is whether a limit is a cap. A limit $L$ is only a ceiling if the function never goes above it. You can check this with an inequality. For example, $\dfrac{2x}{x + 1} < 2$ for $x > 0$ because $2x < 2x + 2$. If a profit model has limit $L$ (in thousands of dollars) and stays below it, the company can never make more than $1000L$ dollars, no matter how much it spends.
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 $0,01$.
The concentration of a certain drug in the blood (in mg/L) $t$ hours after its oral administration follows the function $C(t) = \frac{30t}{t^2 + 9}$. Calculate what…
The concentration of a certain drug in the blood (in mg/L) $t$ hours after its oral administration follows the function $C(t) = \frac{30t}{t^2 + 9}$. Calculate what value the concentration approaches in the long run. Next, set up and solve an inequality to determine after what exact hour the blood concentration will be strictly less than $1$ mg/L.
A software company estimates that its annual profit (in thousands of dollars) as a function of advertising expenses $x$ (also in thousands of dollars) is modeled by the…
A software company estimates that its annual profit (in thousands of dollars) as a function of advertising expenses $x$ (also in thousands of dollars) is modeled by the function $B(x) = \frac{120x^2 - 500}{3x^2 + 2x + 10}$. Is there a maximum limit for profits no matter how much is invested in advertising indefinitely? Justify your answer by calculating the limit at infinity and state how many dollars this cap amounts to.
Back to the pizza: Emma's pizza cools toward 70 °F, the temperature of the kitchen, but it never gets to 0 °F. The horizontal asymptote $y = 70$ is the answer to "what happens if you wait forever?"
Summary
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$\lim_{x\to+\infty} f(x) = L$ means $f(x)$ gets closer and closer to $L$. The line $y = L$ is a horizontal asymptote. A limit of $\pm\infty$ means the values grow without bound, and there is no horizontal asymptote on that side.
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Polynomials: only the leading term $ax^n$ matters. The sign of $a$ and whether $n$ is even or odd decide the two ends.
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Rational functions: compare degrees. Lower degree on top gives $0$. Equal degrees give the ratio of leading coefficients. Higher degree on top gives $\pm\infty$.
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Exponentials: $b > 1$ grows to $\infty$, and $0 < b < 1$ decays to $0$. In $a \cdot b^t + c$ with decay, the asymptote is $y = c$. Exponentials eventually beat any polynomial.
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Piecewise and absolute value: use the rule that applies at each end separately. The two ends can have different asymptotes.
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To ask "how close?", solve $|f(x) - L| < \varepsilon$. Always say what the limit means in the context.