Limit of a sequence
$\dfrac{1}{1}, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \ldots$ Each term is smaller than the previous one, and it gets as close to $0$ as you like, although it never actually equals $0$. That idea of "getting closer forever without running out of patience" is the most important concept in this course: the limit of a sequence. With it you will give a rigorous meaning to things that in 3rd and 4th ESO you only grasped intuitively, such as the sum of infinitely many terms or the value towards which a population settles.
The intuition: getting closer without arriving
Look at $a_n = \dfrac{n+1}{n}$: $2, \ 1{,}5, \ 1{,}333\ldots, \ 1{,}25, \ 1{,}2, \ldots$ The terms get closer and closer to $1$, although none of them is exactly $1$.
Change the general term and observe which value (if any) the bars approach as $n$ increases:
Determine whether the sequence whose general term is $a_n = \frac{(-1)^n \cdot 5}{n}$ is convergent, divergent or oscillating. If it is convergent, state its limit.
Determine whether the sequence whose general term is $a_n = \frac{(-1)^n \cdot 5}{n}$ is convergent, divergent or oscillating. If it is convergent, state its limit.
The formal definition with $\varepsilon$
"Getting as close as you like" can be written rigorously: however small a distance $\varepsilon>0$ you choose, the terms must eventually enter that distance from $L$ and stay inside it.
From a certain position $N$ onwards (which depends on how demanding $\varepsilon$ is), all the following terms fall inside the strip $(L-\varepsilon, L+\varepsilon)$ and never leave it again.
Given the sequence $a_n = \frac{4n - 3}{n + 1}$, we know that its limit is $L = 4$. Using the formal definition of a limit, find from which term $n$ onwards the distance…
Given the sequence $a_n = \frac{4n - 3}{n + 1}$, we know that its limit is $L = 4$. Using the formal definition of a limit, find from which term $n$ onwards the distance between the terms of the sequence and its limit is strictly less than one thousandth, that is, $|a_n - 4| < 0,001$.
Limits of rational sequences
The most common case is a fraction of polynomials. The result depends on comparing the degree of the numerator with that of the denominator.
Find the limit of the rational sequence given by the expression: $c_n = \frac{3n^2 - 5n + 1}{8n^3 + 2n^2 - n}$
Find the limit of the rational sequence given by the expression:
$c_n = \frac{3n^2 - 5n + 1}{8n^3 + 2n^2 - n}$
Calculate the limit of the following sequence, first identifying the degrees of the numerator and denominator: $d_n = \frac{6n^2 - 3n + 5}{2n^2 + 4n - 1}$
Calculate the limit of the following sequence, first identifying the degrees of the numerator and denominator:
$d_n = \frac{6n^2 - 3n + 5}{2n^2 + 4n - 1}$
The fish population of a sustainable fish farm grows according to the mathematical model $P_n = \frac{150000n}{3n + 250}$, where $n$ represents the number of days since…
The fish population of a sustainable fish farm grows according to the mathematical model $P_n = \frac{150000n}{3n + 250}$, where $n$ represents the number of days since the introduction of the fingerlings. Determine the carrying capacity of the environment, that is, the long-term limit of the fish population.
Indeterminate forms with radicals: $\infty - \infty$
When a subtraction of two quantities that both tend to infinity appears, multiplying and dividing by the conjugate turns the subtraction into a fraction.
Calculate the limit of the sequence involving a simple square root in the numerator: $h_n = \frac{\sqrt{16n^2 + 3n}}{2n + 5}$
Calculate the limit of the sequence involving a simple square root in the numerator:
$h_n = \frac{\sqrt{16n^2 + 3n}}{2n + 5}$
Calculate the limit of the following irrational sequence, multiplying and dividing by the conjugate expression to resolve the indeterminate form:…
Calculate the limit of the following irrational sequence, multiplying and dividing by the conjugate expression to resolve the indeterminate form:
$\lim_{n \to \infty} \left( \sqrt{n^2 + 6n} - n \right)$
Convergent, divergent or oscillating
Not every sequence has a limit. There are three possible behaviours as $n\to\infty$.
Calculate the limit of the following polynomial sequence as $n$ tends to infinity: $b_n = -2n^3 + 7n^2 - 4n + 12$
Calculate the limit of the following polynomial sequence as $n$ tends to infinity:
$b_n = -2n^3 + 7n^2 - 4n + 12$
The temperature in degrees Celsius of a piece of metal cooling in a room follows the sequence $T_n = 22 + 85 \cdot (0,82)^n$, where $n$ is the time elapsed in minutes.…
The temperature in degrees Celsius of a piece of metal cooling in a room follows the sequence $T_n = 22 + 85 \cdot (0,82)^n$, where $n$ is the time elapsed in minutes. Calculate the limit of the sequence $T_n$ as $n \to \infty$ and interpret the result in the context of the problem.