Limit of a Sequence
Yaritza drinks half of a passion fruit juice. Then she drinks half of what's left, and then half of what's left again. What remains in the glass is $\dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}, \ldots$ Will the juice ever run out? Each term is smaller than the one before and gets as close to $0$ as you like, even though it never equals exactly $0$. That idea is the limit of a sequence, and it is the foundation of all the calculus that comes next: derivatives and integrals.
The idea of a limit
Look at $a_n = \dfrac{n+1}{n}$: $2,\ 1.5,\ 1.33\ldots,\ 1.25,\ 1.2,\ \ldots$ The terms get closer and closer to $1$, although none of them equals exactly $1$.
Change the general term and watch which value (if any) the bars approach as $n$ increases:
There are three possible behaviors when $n\to\infty$:
Determine whether the sequence whose general term is $a_n = \frac{(-1)^n \times 5}{n}$ is convergent, divergent or oscillating. If it is convergent, state its limit.
Determine whether the sequence whose general term is $a_n = \frac{(-1)^n \times 5}{n}$ is convergent, divergent or oscillating. If it is convergent, state its limit.
Rational and polynomial sequences
The most common case is a quotient of polynomials in $n$. For a single polynomial, the highest-degree term dominates: for example, in $-2n^3+7n^2-4n+12$ the term $-2n^3$ dominates, and it tends to $-\infty$.
For a quotient, 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:
$c_n = \frac{3n^2 - 5n + 1}{8n^3 + 2n^2 - n}$
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.
Find the limit of the following sequence, first expanding the algebraic identities in the numerator to identify the terms of highest degree:…
Find the limit of the following sequence, first expanding the algebraic identities in the numerator to identify the terms of highest degree:
$\lim_{n \to \infty} \frac{(3n+2)^2 - (3n-2)^2}{n+5}$
Sequences with $r^n$
Let's go back to the juice: $\left(\dfrac{1}{2}\right)^n$. Multiplying many times by a number less than $1$ makes things smaller; by a number greater than $1$, it makes them bigger.
A very common use is the temperature of something that is cooling down. In $T_n = 22 + 85\cdot(0.82)^n$, since $0.82^n \to 0$, we get $T_n \to 22$: the metal ends up at room temperature.
Another application is adding infinitely many terms of a geometric progression. The sum of the first $n$ terms is $S_n = a_1\dfrac{1-r^n}{1-r}$. If $|r|<1$, then $r^n\to 0$ and the sum settles down.
Study the behavior and calculate the limit of the exponential-type sequence: $f_n = 12 \times \left(\frac{3}{4}\right)^n$
Study the behavior and calculate the limit of the exponential-type sequence:
$f_n = 12 \times \left(\frac{3}{4}\right)^n$
The temperature in degrees Celsius of a piece of metal cooling in a room follows the sequence $T_n = 22 + 85 \times (0,82)^n$, 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 $T_n = 22 + 85 \times (0,82)^n$, 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.
Starting from a square of area $A_1 = 400 \text{ cm}^2$, a sequence of squares is constructed by joining the midpoints of the sides of the previous square, as shown in…
Starting from a square of area $A_1 = 400 \text{ cm}^2$, a sequence of squares is constructed by joining the midpoints of the sides of the previous square, as shown in the figure. Given that the area of each new square is half the area of the previous square, calculate the limit of the sum of the areas of the infinitely many squares thus constructed: $S_\infty = \lim_{n \to \infty} (A_1 + A_2 + \dots + A_n)$.
Indeterminate forms with radicals: $\infty - \infty$
When you have a difference of two quantities that both tend to infinity, multiplying and dividing by the conjugate turns the subtraction into a fraction.
If the radical appears only in a fraction, as in $\dfrac{\sqrt{16n^2+3n}}{2n+5}$, it is enough to see that $\sqrt{16n^2+3n}$ behaves like $4n$: the limit is $\dfrac{4}{2} = 2$.
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:
$\lim_{n \to \infty} \left( \sqrt{n^2 + 6n} - n \right)$
Calculate the limit of the following sequence, which features a double square root and requires careful application of the conjugate technique:…
Calculate the limit of the following sequence, which features a double square root and requires careful application of the conjugate technique:
$\lim_{n \to \infty} n \times \left( \sqrt{n^2 + 4} - n \right)$
Extension: sequences defined by recurrence
Sometimes each term is obtained from the previous one. If we know the sequence converges to $L$, then $a_n$ and $a_{n+1}$ both tend to the same $L$. We substitute $L$ for both and solve the equation.
A patient receives a daily dose of a medication. After absorption and elimination throughout the day, the concentration of the drug in the blood just before the next…
A patient receives a daily dose of a medication. After absorption and elimination throughout the day, the concentration of the drug in the blood just before the next dose follows the recurrence relation $C_n = 15 + C_{n-1} \times 0.4$, where $15 \text{ mg/L}$ is the contribution of the new dose and the factor $0.4$ indicates that $40\%$ of the previous day's drug remains. If the initial concentration is $C_1 = 15$, calculate the long-term equilibrium concentration in the blood, that is, $\lim_{n \to \infty} C_n$.
A sequence is defined by the following recurrence relation: the first term is $a_1 = \sqrt{6}$, and each subsequent term is obtained as $a_{n+1} = \sqrt{6 + a_n}$.…
A sequence is defined by the following recurrence relation: the first term is $a_1 = \sqrt{6}$, and each subsequent term is obtained as $a_{n+1} = \sqrt{6 + a_n}$. Assuming that the sequence is increasing and bounded above (and is therefore convergent and has a limit $L$), set up the equation that the limit must satisfy and find it.
Extension: parameters in a limit with radicals
If $\sqrt{9n^2+an+2}-bn$ has a finite limit, the highest-degree terms must cancel. That forces $b=3$ (the square root of the coefficient $9$). With $b=3$, the conjugate gives $\dfrac{an+2}{\sqrt{9n^2+an+2}+3n} \to \dfrac{a}{6}$, and we set that equal to the requested value.
Find the values of the real parameters $a$ and $b$ (with $b > 0$) such that the following equality holds:…
Find the values of the real parameters $a$ and $b$ (with $b > 0$) such that the following equality holds:
$\lim_{n \to \infty} \left( \sqrt{9n^2 + an + 2} - bn \right) = 4$
Extension: the number $e$ and the indeterminate form $1^\infty$
If the base tends to $1$ and the exponent to infinity, we cannot say that the result is $1$. With $\left(1+\dfrac{1}{n}\right)^n$ something curious happens: the base gets close to $1$ but the exponent grows, and the two effects balance out at a fixed number.
This lets us compute similar limits. For example, $\left(1+\dfrac{1}{n}\right)^{7n} = \left[\left(1+\dfrac{1}{n}\right)^n\right]^7 \to e^7$.
One application is continuous compounding. If $\text{RD}\$100{,}000$ is deposited at $4\%$ annual interest for $5$ years and the interest is compounded $n$ times per year, the capital is $Cn = 100{,}000\left(1+\dfrac{0.04}{n}\right)^{5n}$. When $n\to\infty$, the exponent gives $e^{0.04\times 5}=e^{0.2}$, and the limiting capital is $100{,}000\,e^{0.2}\approx$ $\text{RD}\$122{,}140$.
Applying the definition of the number $e$, calculate the limit of the following sequence: $g_n = \left(1 + \frac{1}{n}\right)^{7n}$
Applying the definition of the number $e$, calculate the limit of the following sequence:
$g_n = \left(1 + \frac{1}{n}\right)^{7n}$
Calculate the limit of the sequence, transforming the expression to apply the limit definition of the number $e$:…
Calculate the limit of the sequence, transforming the expression to apply the limit definition of the number $e$:
$\lim_{n \to \infty} \left( \frac{n + 7}{n + 2} \right)^{4n}$
Solve the following limit that leads to an indeterminate form of the type $1^\infty$, paying special attention to the algebraic operations in the exponent:…
Solve the following limit that leads to an indeterminate form of the type $1^\infty$, paying special attention to the algebraic operations in the exponent:
$\lim_{n \to \infty} \left( \frac{n^2 - n + 3}{n^2 + n + 1} \right)^{n^2 - 1}$
Limits and growth without bound
A sequence can grow without stopping if it is multiplied by a factor greater than $1$ at each step. This is the case of the Koch curve: it starts from a segment of $1$ m, and at each step its length is multiplied by $\frac{4}{3}$. The length at step $n$ is $L_n = \left(\frac{4}{3}\right)^n$. Since $r=\frac{4}{3}>1$, $L_n\to+\infty$: the curve has infinite length even though it fits in a region of the plane of finite size.
The figure shows the first three iterations in the construction of the Koch curve. If we start from an initial segment of length $L_0 = 1$ meter (panel A), at each step…
The figure shows the first three iterations in the construction of the Koch curve. If we start from an initial segment of length $L_0 = 1$ meter (panel A), at each step $n$ the total length of the curve is multiplied by $\frac{4}{3}$ compared to the previous step. Write the expression for the length $L_n$ at step $n$ and calculate $\lim_{n \to \infty} L_n$. What geometric conclusion can be drawn from this result?
Extension: the definition with $\varepsilon$
"Getting as close as we want" is written rigorously like this: no matter how small a distance $\varepsilon>0$ you choose, the terms must eventually enter and stay within that distance of $L$.
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, $|a_n - 4| < 0.001$.