Exercises: Operations with sequences. Indeterminate forms

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 $b_n = \frac{7n^2 - n}{n^3 + 4n - 1}$.

Find the limit of the sequence $c_n = \frac{-2n^4 + 3n^2}{8n^2 + 5n - 2}$.

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 exponential-type sequence $e_n = \left( 1 + \frac{1}{n} \right)^{3n}$.

Solve the following limit involving radicals by multiplying and dividing by the conjugate expression:

$\lim_{n \to \infty} \left( \sqrt{n^2 + 5n} - n \right)$

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 $\lim_{n \to \infty} (a_n \times b_n)$.

Find the limit of the sequence $f_n = \left( \frac{n+4}{n} \right)^n$.

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 in thousands. What value does the channel's audience stabilise towards in the very long term?

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.

Identify the indeterminate form and calculate the limit of the sequence $a_n = \left( \frac{3n^2 + 5}{3n^2 - 2} \right)^{4n^2}$.

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.

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 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.

Calculate the limit of the difference of algebraic fractions:

$\lim_{n \to \infty} \left( \frac{n^3 + 2n^2}{n^2 - 1} - \frac{n^2 + 3}{n + 1} \right)$

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 $n \to \infty$.

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:

$\lim_{n \to \infty} \left( \frac{2n^2 + 3n}{n + 1} - an - b \right) = 0$

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 $a$?

Solve the following limit that combines the sum of the first $n$ terms of an arithmetic progression and a polynomial indeterminate form:

$\lim_{n \to \infty} \frac{1 + 2 + 3 + \dots + n}{3n^2 + 2n - 1}$

Resolve the indeterminate form $\infty - \infty$ generated by the difference of two square roots of quadratic polynomials:

$\lim_{n \to \infty} \left( \sqrt{n^2 + 5n + 1} - \sqrt{n^2 - 3n + 2} \right)$

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.

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 $n \to \infty$.

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 $n \to \infty$.