Operations with sequences. Indeterminate forms

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

Design the general term of a rational sequence $a_n = \frac{p\times n + q}{r\times n + s}$ (with $p, q, r, s \in \mathbb{R}$ and $r \neq 0$) that simultaneously satisfies the following three conditions:

1) Its limit at infinity is exactly $4$: $\lim_{n \to \infty} a_n = 4$.

2) Its first term is $a_1 = 1$.

3) The sequence is strictly increasing for all $n \ge 1$ (that is, $a_{n+1} > a_n$).

Explain the step-by-step derivation of the values you choose for the coefficients and algebraically prove that the sequence you proposed is strictly increasing for all $n \ge 1$.

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 $c_n = \frac{-2n^4 + 3n^2}{8n^2 + 5n - 2}$.

In a Leaving Certificate Mathematics exam, Seán has to calculate the limit $\lim_{n \to \infty} \left( \frac{2n^2 + 3}{n} - 2n \right)$. He writes on his paper: "Since $\lim \frac{2n^2+3}{n} = \infty$ and $\lim 2n = \infty$, then the result is $\infty - \infty = 0$".

1. a) Explain with reasons why Seán's argument is mathematically incorrect and what it actually means for $\infty - \infty$ to be an indeterminate form.
2. b) Solve the limit rigorously by transforming the expression using a common denominator.

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:

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

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 $n \to \infty$.

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 number of iterations tends to infinity.

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

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

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 $a_n = \left( \frac{3n^2 + 5}{3n^2 - 2} \right)^{4n^2}$.