Operations with Sequences. Indeterminate Forms
A video channel run by a young person from Santiago gains followers every month according to a formula, and they want to know whether it will ever level off. That is calculating the limit of a sequence. But when the formula mixes several sequences that grow without bound, the result is not obvious: indeterminate forms appear, and we need a technique to resolve them.
Algebra of Limits
These rules also hold when one of the limits is infinite, as long as the result is determined:
| Operation | Result |
|---|---|
| $\infty+\infty$ | $+\infty$ |
| $k\times\infty$ with $k>0$ | $+\infty$ |
| $\dfrac{k}{\infty}$ | $0$ |
| $\dfrac{k}{0^+}$ with $k>0$ | $+\infty$ |
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 $\lim_{n \to \infty} (a_n \times b_n)$.
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 $\lim_{n \to \infty} (a_n \times b_n)$.
The number of followers of a popular science channel grows according to the sequence $S_n = \frac{1200n^2 + 50n}{3n^2 + 25}$, where $n$ is time in months and $S_n$ is…
The number of followers of a popular science channel grows according to the sequence $S_n = \frac{1200n^2 + 50n}{3n^2 + 25}$, where $n$ is time in months and $S_n$ is the number of followers in thousands. What value does the channel's audience stabilise towards in the very long term?
The Catalog of Indeterminate Forms
Resolving $\frac{\infty}{\infty}$: Comparing Degrees
For a quotient of polynomials in $n$, you divide by the highest power of $n$ or, equivalently, compare the degrees.
When there are powers of binomials, you expand first, because sometimes the leading terms cancel.
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 $b_n = \frac{7n^2 - n}{n^3 + 4n - 1}$.
A financial analyst studies the yield 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 yield 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 limiting yield by resolving the corresponding indeterminate form.
Factoring and the $\frac{0}{0}$ Form
Sometimes the numerator and denominator share a factor. You factor, cancel, and the indeterminate form disappears.
A sequence can have the form $\frac{0}{0}$ when the numerator and denominator both tend to $0$. You multiply the top and bottom by the power of $n$ that eliminates the denominators.
Other Quotients: Higher Degrees, Sums, and Exponentials
If the numerator has a higher degree than the denominator, the limit is infinite and you only need to find the sign.
If the numerator is a sum of terms that depend on $n$, you first calculate the sum. Remember that $1+2+\dots+n=\dfrac{n(n+1)}{2}$ (sum of an arithmetic progression).
When there are powers with $n$ in the exponent, you divide by the one that grows fastest: $3^n$ beats $2^n$, and $5^n$ beats $3^n$. The quotients $\left(\frac{2}{3}\right)^n$ tend to $0$.
Find the limit of the sequence $c_n = \frac{-2n^4 + 3n^2}{8n^2 + 5n - 2}$.
Find the limit of the sequence $c_n = \frac{-2n^4 + 3n^2}{8n^2 + 5n - 2}$.
Solve the following limit that combines the sum of the first $n$ terms of an arithmetic progression and a polynomial indeterminate form:…
Solve the following limit that combines the sum of the first $n$ terms of an arithmetic progression and a polynomial indeterminate form:
$\lim_{n \to \infty} \frac{1 + 2 + 3 + \dots + n}{3n^2 + 2n - 1}$
The cumulative 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 inflation as…
The cumulative 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 inflation as $n \to \infty$.
Resolving $\infty-\infty$ with Fractions
If the subtraction involves algebraic fractions, you find a common denominator, simplify the numerator, and are left with a quotient of polynomials.
Resolve the indeterminate form $\infty - \infty$ in the following sequence by first simplifying 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 simplifying 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:
$\lim_{n \to \infty} \left( \frac{n^3 + 2n^2}{n^2 - 1} - \frac{n^2 + 3}{n + 1} \right)$
Resolving $\infty-\infty$ with Radicals: The Conjugate
If the subtraction contains a root, you multiply and divide by the conjugate. This produces the difference of squares $(A-B)(A+B)=A^2-B^2$ and the root disappears from the numerator.
If the root is in a quotient, it is enough to divide the entire expression by $n$.
Solve the following limit involving radicals by multiplying and dividing by the conjugate expression: $\lim_{n \to \infty} \left( \sqrt{n^2 + 5n} - n \right)$
Solve the following limit involving radicals by multiplying and dividing by the conjugate expression:
$\lim_{n \to \infty} \left( \sqrt{n^2 + 5n} - n \right)$
The deviation in millimeters of a robotic arm in iteration $n$ is given by the sequence $d_n = \sqrt{9n^2 + 4n} - 3n$. Calculate the limiting deviation error as the…
The deviation in millimeters of a robotic arm in iteration $n$ is given by the sequence $d_n = \sqrt{9n^2 + 4n} - 3n$. Calculate the limiting deviation error as the number of iterations tends to infinity.
Resolve the indeterminate form in the sequence with nested radicals in the numerator and denominator:…
Resolve the indeterminate form in the sequence with nested radicals in the numerator and denominator:
$\lim_{n \to \infty} \frac{\sqrt{16n^2 + 3n} - 2n}{5n + \sqrt{n^2 - 1}}$
Two Roots and a Root Multiplied by $n$
When you subtract two roots, the conjugate is their sum.
If the difference is multiplied by $n$ (an $\infty\times0$ indeterminate form), you resolve it the same way: conjugate first, then divide.
The net migration rate of a region is estimated according to the sequence $I_n = n \left( \sqrt{n^2 + 2} - \sqrt{n^2 - 2} \right)$. Determine the limiting trend of this…
The net migration rate of a region is estimated according to the sequence $I_n = n \left( \sqrt{n^2 + 2} - \sqrt{n^2 - 2} \right)$. Determine the limiting trend of this rate as $n \to \infty$.
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:
$\lim_{n \to \infty} \left( \sqrt{n^2 + 5n + 1} - \sqrt{n^2 - 3n + 2} \right)$
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 Indeterminate Form $1^{\infty}$
When the base tends to $1$ and the exponent to infinity, the result is not automatically $1$. For example, consider a savings account with compound interest: the base is just barely above $1$, but it is raised to many periods.
Calculate the limit of the exponential-type sequence $e_n = \left( 1 + \frac{1}{n} \right)^{3n}$.
Calculate the limit of the exponential-type sequence $e_n = \left( 1 + \frac{1}{n} \right)^{3n}$.
Find the limit of the sequence $f_n = \left( \frac{n+4}{n} \right)^n$.
Find the limit of the sequence $f_n = \left( \frac{n+4}{n} \right)^n$.
$1^\infty$ with More Complicated Bases
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 during 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 during an iterative cooling process follows the sequence $T_n = \left( \frac{n^2 - n}{n^2 + 3} \right)^{2n+1}$ in degrees Celsius above ambient temperature. Determine the limit of this temperature.
Show, by calculating the limit, 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 towards.
Show, by calculating the limit, 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 towards.
Finding Parameters from a Limit
Find the value of the parameter $k \in \mathbb{R}$ such that the limit of the sequence $a_n = \sqrt{4n^2 + kn} - 2n$ is equal to $5$.
Find the value of the parameter $k \in \mathbb{R}$ such that the limit of the sequence $a_n = \sqrt{4n^2 + kn} - 2n$ is equal to $5$.
Determine the values of the parameters $a$ and $b$ such that the equality holds: $\lim_{n \to \infty} \left( \frac{2n^2 + 3n}{n + 1} - an - b \right) = 0$
Determine the values of the parameters $a$ and $b$ such that the equality holds:
$\lim_{n \to \infty} \left( \frac{2n^2 + 3n}{n + 1} - an - b \right) = 0$
In a crystallisation process, the concentration of a reactant follows the sequence $C_n = \left( \frac{n + a}{n + 2} \right)^{3n}$. If the long-term limiting…
In a crystallisation process, the concentration of a reactant follows the sequence $C_n = \left( \frac{n + a}{n + 2} \right)^{3n}$. If the long-term limiting concentration is known to be $e^6$, what must be the value of the parameter $a$?
Summary
Back to the channel from the beginning: a formula like $\frac{1200n^2+50n}{3n^2+25}$ is a quotient with equal degrees, and the number of followers levels off at $\frac{1200}{3}=400$ (thousand).