Operations with Sequences and Indeterminate Forms
Aaliyah gains $n^2 + 5n$ new followers by day $n$, and Marcus gains $3n^2 + 1$. Both totals grow without bound. In the long run, how do their follower counts compare? Each one is "going to infinity," so you can't just plug in $\infty$. In this section you learn how to combine sequences, when the limit behaves exactly as you expect, and what to do when it doesn't.
Limit Laws and Infinity Arithmetic
You already know how to find the limit of a single sequence. Now you will add, multiply and divide sequences. When the limits are finite numbers, everything works the way you expect.
Arithmetic with infinity
When one of the sequences grows without bound, some results are still automatic. Here $k$ is a finite number.
| Expression | Result |
|---|---|
| $\infty + \infty$ | $\infty$ |
| $\infty \cdot \infty$ | $\infty$ |
| $k \cdot \infty$ | $+\infty$ if $k > 0$, $-\infty$ if $k < 0$ |
| $\dfrac{k}{\infty}$ | $0$ |
| $\dfrac{k}{0^+}$ (with $k>0$) | $+\infty$ |
The indeterminate forms
Some combinations have no automatic answer. These are the indeterminate forms:
$$\frac{\infty}{\infty} \qquad \infty - \infty \qquad 0 \cdot \infty \qquad 1^{\infty} \qquad \frac{0}{0}$$"Indeterminate" means the form alone doesn't tell you the limit. Different sequences with the same form give different limits. You must rewrite the expression first, and the rest of this page is a toolbox for doing that.
Given the sequences $a_n = \frac{2n}{n+1}$ and $b_n = 3n^2$, calculate the limits $\lim_{n \to \infty} \frac{a_n}{b_n}$ and $\lim_{n \to \infty} (a_n \cdot b_n)$.
Given the sequences $a_n = \frac{2n}{n+1}$ and $b_n = 3n^2$, calculate the limits $\lim_{n \to \infty} \frac{a_n}{b_n}$ and $\lim_{n \to \infty} (a_n \cdot b_n)$.
$\frac{\infty}{\infty}$: Compare Degrees
The most common indeterminate form is a quotient of two polynomials in $n$. Both go to infinity, but one can win the race.
Look at $\dfrac{6n^2 - n}{2n^2 + 7}$. Divide the numerator and denominator by the highest power in the denominator, $n^2$:
$$\frac{6 - \frac{1}{n}}{2 + \frac{7}{n^2}} \to \frac{6 - 0}{2 + 0} = 3$$Every term with $n$ in the denominator goes to $0$, so only the leading terms survive.
Back to the opening: Aaliyah has $n^2 + 5n$ followers and Marcus has $3n^2 + 1$. The degrees tie, so the ratio is $\dfrac{n^2 + 5n}{3n^2 + 1} \to \dfrac{1}{3}$. In the long run, Aaliyah has about one third as many followers as Marcus. Both are infinite, yet the ratio settles at $\frac13$.
Calculate the limit of the sequence $a_n = \frac{4n^3 - 2n + 1}{5n^3 + n^2 - 3}$.
Calculate the limit of the sequence $a_n = \frac{4n^3 - 2n + 1}{5n^3 + n^2 - 3}$.
Determine the value of the limit of the sequence $b_n = \frac{7n^2 - n}{n^3 + 4n - 1}$.
Determine the value of the limit of the sequence $b_n = \frac{7n^2 - n}{n^3 + 4n - 1}$.
Find the limit of the sequence $c_n = \frac{-2n^4 + 3n^2}{8n^2 + 5n - 2}$.
Find the limit of the sequence $c_n = \frac{-2n^4 + 3n^2}{8n^2 + 5n - 2}$.
Expand First, Then Compare
Sometimes the numerator or denominator is not written as a simple polynomial. You may need to expand it or add up a sum before you can see the true degree.
A biologist models the growth of a population of bacteria in a closed culture using the sequence $P_n = \frac{1200n^2 + 50n}{3n^2 + 25}$, where $n$ is the time in hours…
A biologist models the growth of a population of bacteria in a closed culture using the sequence $P_n = \frac{1200n^2 + 50n}{3n^2 + 25}$, where $n$ is the time in hours and $P_n$ is the number of individuals in thousands. Towards what value does the population stabilise in the very long term?
A financial analyst studies the performance of an investment fund whose rate of return in month $n$ follows the sequence $R_n = \frac{(2n+1)^3 - 8n^3}{3n^2 + 5n - 1}$.…
A financial analyst studies the performance of an investment fund whose rate of return in month $n$ follows the sequence $R_n = \frac{(2n+1)^3 - 8n^3}{3n^2 + 5n - 1}$. Determine the limit return by resolving the corresponding indeterminate form.
Resolve the following limit that combines the sum of the first $n$ terms of an arithmetic progression and a polynomial indeterminate form:…
Resolve the following limit that combines the sum of the first $n$ terms of an arithmetic progression and a polynomial indeterminate form:
$\lim_{n \to \infty} \frac{1 + 2 + 3 + \dots + n}{3n^2 + 2n - 1}$
Radicals and Exponentials: Divide by the Dominant Term
Radicals and exponentials also have a "dominant" part. A radical like $\sqrt{9n^2 + n}$ acts like $3n$ (degree $1$). An exponential beats any power of $n$. Divide the top and bottom by the dominant term, and the rest goes to $0$.
Resolve the indeterminate form in the sequence with iterated radicals in the numerator and denominator:…
Resolve the indeterminate form in the sequence with iterated radicals in the numerator and denominator:
$\lim_{n \to \infty} \frac{\sqrt{16n^2 + 3n} - 2n}{5n + \sqrt{n^2 - 1}}$
The accumulated inflation in a theoretical macroeconomic model is given by the sequence $I_n = \frac{3^n + 5^{n+1}}{5^n + 2^{n+2}}$. Determine the limit of the inflation…
The accumulated inflation in a theoretical macroeconomic model is given by the sequence $I_n = \frac{3^n + 5^{n+1}}{5^n + 2^{n+2}}$. Determine the limit of the inflation as $n \to \infty$.
$\infty - \infty$ with Rational Expressions
When you subtract two algebraic fractions that both blow up, combine them over a common denominator and simplify. The indeterminate form usually disappears.
Finding parameters
Sometimes the limit is given and you must find an unknown constant. Combine the fractions, then force the degree of the numerator to be low enough.
Resolve the indeterminate form $\infty - \infty$ in the following sequence by first operating on the algebraic fractions: $d_n = \frac{n^2 + 1}{n} - \frac{n^2 - 2}{n+1}$
Resolve the indeterminate form $\infty - \infty$ in the following sequence by first operating on the algebraic fractions:
$d_n = \frac{n^2 + 1}{n} - \frac{n^2 - 2}{n+1}$
Calculate the limit of the difference of algebraic fractions: $\lim_{n \to \infty} \left( \frac{n^3 + 2n^2}{n^2 - 1} - \frac{n^2 + 3}{n + 1} \right)$
Calculate the limit of the difference of algebraic fractions:
$\lim_{n \to \infty} \left( \frac{n^3 + 2n^2}{n^2 - 1} - \frac{n^2 + 3}{n + 1} \right)$
Determine the values of the parameters $a$ and $b$ so that the equality is satisfied: $\lim_{n \to \infty} \left( \frac{2n^2 + 3n}{n + 1} - an - b \right) = 0$
Determine the values of the parameters $a$ and $b$ so that the equality is satisfied:
$\lim_{n \to \infty} \left( \frac{2n^2 + 3n}{n + 1} - an - b \right) = 0$
$\infty - \infty$ with a Radical: The Conjugate
When the difference involves a square root, multiply and divide by the conjugate. This uses the difference of squares, $(A - B)(A + B) = A^2 - B^2$, which makes the root disappear.
Resolve the following limit with radicals by multiplying and dividing by the conjugate expression: $\lim_{n \to \infty} \left( \sqrt{n^2 + 5n} - n \right)$
Resolve the following limit with radicals by multiplying and dividing by the conjugate expression:
$\lim_{n \to \infty} \left( \sqrt{n^2 + 5n} - n \right)$
The deviation in inches of a robotic arm in its iteration $n$ is given by the sequence $d_n = \sqrt{9n^2 + 4n} - 3n$. Calculate the limit deviation error as the number…
The deviation in inches of a robotic arm in its iteration $n$ is given by the sequence $d_n = \sqrt{9n^2 + 4n} - 3n$. Calculate the limit deviation error as the number of iterations tends to infinity.
Find the value of the parameter $k \in \mathbb{R}$ so that the limit of the sequence $a_n = \sqrt{4n^2 + kn} - 2n$ is equal to $5$.
Find the value of the parameter $k \in \mathbb{R}$ so that the limit of the sequence $a_n = \sqrt{4n^2 + kn} - 2n$ is equal to $5$.
$\infty - \infty$ with Two Radicals
If both terms are square roots, use the same idea. The conjugate of $\sqrt{A} - \sqrt{B}$ is $\sqrt{A} + \sqrt{B}$, and the product is $A - B$. The roots vanish from the numerator and what is left is a quotient of degree-$1$ pieces.
Resolve the indeterminate form $\infty - \infty$ generated by the difference of two square roots of quadratic polynomials:…
Resolve the indeterminate form $\infty - \infty$ generated by the difference of two square roots of quadratic polynomials:
$\lim_{n \to \infty} \left( \sqrt{n^2 + 5n + 1} - \sqrt{n^2 - 3n + 2} \right)$
$0 \cdot \infty$: Turn the Product into a Fraction
A product where one factor goes to $0$ and the other to $\infty$ is also indeterminate. Often the factor that goes to $0$ is a difference of radicals, and the conjugate converts the product into a quotient you already know how to handle.
The net migration index of a region is estimated according to the sequence $I_n = n \left( \sqrt{n^2 + 2} - \sqrt{n^2 - 2} \right)$. Determine the limit trend of this…
The net migration index of a region is estimated according to the sequence $I_n = n \left( \sqrt{n^2 + 2} - \sqrt{n^2 - 2} \right)$. Determine the limit trend of this index as $n \to \infty$.
In the study of relativistic kinetic energy at low speeds, a mathematical approximation involving the sequence $E_n = n^2 \left( \sqrt{1 + \frac{2}{n^2}} - 1 \right)$ is…
In the study of relativistic kinetic energy at low speeds, a mathematical approximation involving the sequence $E_n = n^2 \left( \sqrt{1 + \frac{2}{n^2}} - 1 \right)$ is used. Determine the value of the limit of the sequence $E_n$ as $n \to \infty$.
The $1^{\infty}$ Form and the Number $e$
Suppose a bank offers 100% yearly interest on $\$1$, but compounds it $n$ times a year. After one year you have $\left(1 + \frac{1}{n}\right)^n$ dollars. The base tends to $1$ and the exponent tends to $\infty$. Is the result $1$? Is it $\infty$? Neither.
The values increase but level off. The limit is a famous irrational number.
So even if the bank compounds every second, $\$1$ grows to only about $\$2.72$ in a year. More generally, if $m_n \to \infty$ then $\left(1 + \frac{1}{m_n}\right)^{m_n} \to e$. Use that to find limits where the exponent is a multiple of the denominator.
Calculate the limit of the exponential sequence $e_n = \left( 1 + \frac{1}{n} \right)^{3n}$.
Calculate the limit of the exponential sequence $e_n = \left( 1 + \frac{1}{n} \right)^{3n}$.
The General Rule for $1^{\infty}$
Most $1^{\infty}$ limits don't look like $\left(1 + \frac{1}{n}\right)^n$. Here is a rule that handles all of them.
Find the limit of the sequence $f_n = \left( \frac{n+4}{n} \right)^n$.
Find the limit of the sequence $f_n = \left( \frac{n+4}{n} \right)^n$.
Identify the indeterminate form and calculate the limit of the sequence $a_n = \left( \frac{3n^2 + 5}{3n^2 - 2} \right)^{4n^2}$.
Identify the indeterminate form and calculate the limit of the sequence $a_n = \left( \frac{3n^2 + 5}{3n^2 - 2} \right)^{4n^2}$.
The temperature of an electronic component in an iterative cooling process follows the sequence $T_n = \left( \frac{n^2 - n}{n^2 + 3} \right)^{2n+1}$ in degrees…
The temperature of an electronic component in an iterative cooling process follows the sequence $T_n = \left( \frac{n^2 - n}{n^2 + 3} \right)^{2n+1}$ in degrees Fahrenheit above room temperature. Determine the limit of this temperature.
$1^{\infty}$ with Parameters and Complicated Exponents
The rule is flexible. The exponent doesn't have to be a multiple of $n$, and the base can contain an unknown constant that you solve for.
In a crystallisation process, the concentration of a reactant follows the sequence $C_n = \left( \frac{n + a}{n + 2} \right)^{3n}$. If it is known that the limit…
In a crystallisation process, the concentration of a reactant follows the sequence $C_n = \left( \frac{n + a}{n + 2} \right)^{3n}$. If it is known that the limit concentration in the long term is $e^6$, what must the value of the parameter $a$ be?
Prove, by calculating the limit, towards what value the sequence defined by $a_n = \left( \frac{n^2 + 5n - 1}{n^2 + 2n + 3} \right)^{\frac{n^2+1}{n}}$ tends.
Prove, by calculating the limit, towards what value the sequence defined by $a_n = \left( \frac{n^2 + 5n - 1}{n^2 + 2n + 3} \right)^{\frac{n^2+1}{n}}$ tends.