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 $f_n = \left( \frac{n+4}{n} \right)^n$.

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.

Given the sequence with general term $a_n = \frac{6n^2 + 50n}{2n^2 - 7}$:

1. a) Calculate mentally or with the help of a calculator the terms $a_{10}$, $a_{100}$, and $a_{1000}$ (rounding to two decimal places) and, from them, make a reasoned estimate about the value toward which the sequence tends as $n \to \infty$.
2. b) Formally calculate $\lim_{n \to \infty} a_n$ by applying the operational rules for quotients of polynomials (dividing numerator and denominator by the highest power of $n$).
3. c) Compare your initial estimate with the exact result obtained and reflect on the influence of the linear term $50n$ as $n$ becomes arbitrarily large.

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 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.

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 $n \to \infty$.

The Central Statistics Office (CSO) periodically publishes indicators on urban sustainability. In a district of Cork that has implemented an intensive electronic waste recycling plan, the percentage of devices properly collected after $n$ years of the campaign is modeled by the sequence:

1. a) Determine the estimated collection percentage for the first year ($n = 1$) and for the fourth year ($n = 4$).
2. b) Calculate $\lim_{n \to \infty} P_n$, detailing each step in resolving the indeterminate form $\frac{\infty}{\infty}$.
3. c) The European Union sets a target for municipalities to continuously exceed $80\,\%$ of selective collection. Based on the value of the limit obtained, interpret whether this district will succeed in stabilizing above this goal in the long term or if the plan will prove insufficient.

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 $n \ge 1$.

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 $n \to \infty$.

To solve the limit $\lim_{n \to \infty} \left(\sqrt{n^2 + 6n} - \sqrt{n^2 - 2n}\right)$, three 5th Year students propose different strategies:

• Aoife (Method A): She factors out $n$ as a common factor, writing $n\left(\sqrt{1 + \frac{6}{n}} - \sqrt{1 - \frac{2}{n}}\right)$, substitutes $\frac{6}{n} \to 0$ and $\frac{2}{n} \to 0$, and concludes that the limit is $\infty \times (1 - 1) = \infty \times 0 = 0$.

• Cian (Method B): He multiplies and divides by the conjugate of the expression to transform the indeterminate form $\infty - \infty$ into a quotient of infinities.

• Niamh (Method C): She completes the square inside each root, observing that $n^2 + 6n = (n+3)^2 - 9$ and that $n^2 - 2n = (n-1)^2 - 1$. She reasons that for huge values of $n$, the numbers $-9$ and $-1$ are negligible compared to the squares, so she approximates the subtraction by $(n+3) - (n-1) = 4$.

Tasks:

1) Explain precisely why Aoife's reasoning is incorrect and at what point she makes a logical error.

2) Fully develop the calculation using Cian's Method B and obtain the exact value of the limit.

3) Compare the formal validity of Cian's method against Niamh's heuristic approximation: can Niamh's proposal be considered a rigorous proof on an exam or merely an intuitive conjecture? Justify your answer.

Consider the family of sequences depending on two real parameters $a, b \in \mathbb{R}$ given by:

1. a) Analyze what type of behavior or indeterminate form $\lim_{n \to \infty} x_n$ exhibits when $a < 0$, when $a = 0$, and when $a > 0$ with $a \neq 3$. Deduce with reasoning what value parameter $a$ must necessarily have so that the limit is a finite real number.
2. b) With that value of $a$ fixed, determine the exact value of parameter $b$ so that the equality $\lim_{n \to \infty} x_n = -1$ holds.
3. c) With the parameters $a$ and $b$ found, analyze whether the terms of the sequence approach $-1$ from above (values greater than $-1$) or from below (values less than $-1$) for large values of $n$. Justify your conclusion by studying the sign of the difference $x_n - (-1)$.