Operations with Sequences. Indeterminate Forms
Regina has a channel with $\frac{n^2+5n}{n+2}$ thousand followers in month $n$, and her friend Mateo has $n$ thousand. Both keep growing without bound, so their difference looks like "$\infty-\infty$". How far apart do they end up in the long run? We'll solve it at the end of the page.
Limit algebra
These rules also work with infinities when the result is determined: $\infty+\infty=+\infty$, $k\times\infty=\pm\infty$ (with $k\neq0$), and $\frac{k}{\infty}=0$.
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 $\lim_{n \to \infty} (a_n \times b_n)$.
Catalog of indeterminate forms
$\frac{\infty}{\infty}$ with polynomials: compare degrees
When $n$ is very large, the highest-degree term dominates a polynomial. That's why, in a quotient of polynomials, only the leading terms matter.
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 before comparing
Sometimes the numerator or denominator comes as powers, products, or long sums. First expand and simplify, then compare degrees.
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?
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.
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:
$\lim_{n \to \infty} \frac{1 + 2 + 3 + \dots + n}{3n^2 + 2n - 1}$
Quotients with exponentials
With powers $a^n$ of different bases, the larger base wins (if $a>1$, $a^n$ grows faster the larger $a$ is). Divide by the power of the largest base.
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 $n \to \infty$.
$\infty-\infty$ with fractions
If both sequences are algebraic fractions, subtract them as fractions: find a common denominator, simplify, and apply the degree criterion.
This answers our opening question: the difference between Regina and Mateo tends to $3$ thousand followers. Both grow without bound, but the gap levels off.
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:
$\lim_{n \to \infty} \left( \frac{n^3 + 2n^2}{n^2 - 1} - \frac{n^2 + 3}{n + 1} \right)$
$\infty-\infty$ with a radical: the conjugate
If a root is subtracted from something, multiply and divide by the conjugate (the same expression with the middle sign changed). Use the special product $(A-B)(A+B)=A^2-B^2$, which eliminates the root.
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.
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 $n \to \infty$.
Two radicals and quotients with roots
If two roots are subtracted, the conjugate is the sum of the two roots. If there are roots in a quotient, first study what each part approaches.
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:
$\lim_{n \to \infty} \frac{\sqrt{16n^2 + 3n} - 2n}{5n + \sqrt{n^2 - 1}}$
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:
$\lim_{n \to \infty} \left( \sqrt{n^2 + 5n + 1} - \sqrt{n^2 - 3n + 2} \right)$
The indeterminate form $1^{\infty}$ and the number $e$
When the base tends to $1$ and the exponent tends to infinity, the result is not $1$: the base is never exactly $1$, and that tiny "extra" gets multiplied many times. This is the situation of savings with compound interest.
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 $f_n = \left( \frac{n+4}{n} \right)^n$.
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 $a_n = \left( \frac{3n^2 + 5}{3n^2 - 2} \right)^{4n^2}$.
$1^{\infty}$ with polynomials and growing exponents
When the base is a quotient of polynomials that tends to $1$ (same degree and same leading coefficient), subtract $1$, simplify, and multiply by the exponent.
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.
Parameters in a limit
When the limit must equal a given value, compute the limit leaving the parameter as a letter and then set it equal to that value.
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 $a$?