Operations with sequences. Indeterminate forms
You already know how to calculate the limit of a single sequence. Now it is time to combine several: adding them, multiplying them, dividing them. Most of the time the limit behaves as you would expect, but there are combinations where the result "disappears" into an indeterminate form, and a technique is needed to uncover it.
Algebra of limits
Calculate the limit of the sequence $a_n = \frac{4n^3 - 2n + 1}{5n^3 + n^2 - 3}$.
Calculate the limit of the sequence $a_n = \frac{4n^3 - 2n + 1}{5n^3 + n^2 - 3}$.
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 $\lim_{n \to \infty} (a_n \cdot b_n)$.
The catalogue of indeterminate forms
When infinities or zeros are involved, some combinations have no automatic result: you have to manipulate the expression before you can apply the algebra of limits.
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 $c_n = \frac{-2n^4 + 3n^2}{8n^2 + 5n - 2}$.
Resolving $\frac{\infty}{\infty}$ with algebraic fractions
When the numerator and denominator are sums or differences of powers, it is best to expand and simplify before applying the degree criterion.
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:
$\lim_{n \to \infty} \frac{1 + 2 + 3 + \dots + n}{3n^2 + 2n - 1}$
Resolving $\infty - \infty$: differences of fractions and radicals
With algebraic fractions, they are added or subtracted like ordinary fractions and then simplified. With radicals, you multiply and divide by the conjugate, just as in the previous section.
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 millimetres 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 millimetres 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 $\infty - \infty$ 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 $1^{\infty}$: a preview
When the base of a power tends to $1$ and the exponent tends to infinity, the result is not automatically $1$: it depends on the "speed" at which the base approaches $1$. This indeterminate form appears constantly with compound interest and populations, and it has its own technique, which you will see in depth (together with the number $e$) in the next section.
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 $e_n = \left( 1 + \frac{1}{n} \right)^{3n}$.
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 $a_n = \left( \frac{3n^2 + 5}{3n^2 - 2} \right)^{4n^2}$.