Limits of Sequences
You pour hot chocolate at $160$ °F into a mug in a $70$ °F kitchen. Minute by minute it cools, but it never quite reaches $70$ °F. What does it mean to say the temperature "approaches" $70$ °F forever? That idea is the limit of a sequence, and it is the foundation of calculus.
What a Limit Means
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 $1$, even though none of them equals $1$.
Change the formula and watch where the bars settle as $n$ grows:
The three possible behaviors
Not every sequence settles down. As $n\to\infty$ there are three possibilities.
Some books simply say "diverges" for both of the last two cases. Here we keep them apart so you can describe exactly what happens.
Here is the toolbox we will use all page long.
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.
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 Formal Definition (ε and N)
"As close as you like" can be made exact. Pick any tolerance $\varepsilon>0$, no matter how tiny. The terms must eventually enter the band from $L-\varepsilon$ to $L+\varepsilon$ and stay there.
Beyond some position $N$ (which depends on how demanding $\varepsilon$ is), every later term lies inside the band and never leaves it.
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 out from which term $n$ it is satisfied…
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 out from which term $n$ it is satisfied that the distance between the terms of the sequence and its limit is strictly less than one thousandth, that is, $|a_n - 4| < 0.001$.
Rational Sequences
The most common case is a fraction of polynomials in $n$. The result depends on comparing the degrees of numerator and 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 the 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 the denominator:
$d_n = \frac{6n^2 - 3n + 5}{2n^2 + 4n - 1}$
Determine the value towards which the sequence $e_n$ tends for very large values of $n$: $e_n = \frac{n^4 + 2n^2 - 7}{5n^2 - 3n + 1}$
Determine the value towards which the sequence $e_n$ tends for very large values of $n$:
$e_n = \frac{n^4 + 2n^2 - 7}{5n^2 - 3n + 1}$
Simplify First: Combining and Expanding
Sometimes the expression does not look like a single fraction. For example, $\dfrac{n^2+3n}{n+1}-n$ is "$\infty-\infty$", which has no automatic answer. Do the algebra first: combine fractions, expand brackets and cancel. Then compare degrees.
A population of bacteria cultured in a laboratory grows according to the mathematical model $P_n = \frac{150000n}{3n + 250}$, where $n$ represents the number of days…
A population of bacteria cultured in a laboratory grows according to the mathematical model $P_n = \frac{150000n}{3n + 250}$, where $n$ represents the number of days since the start of the culture. Determine the carrying capacity of the environment, that is, the limit of the bacteria population in the long term.
Find the limit of the following sequence, first expanding the brackets in the numerator to identify the terms of highest degree:…
Find the limit of the following sequence, first expanding the brackets in the numerator to identify the terms of highest degree:
$\lim_{n \to \infty} \frac{(3n+2)^2 - (3n-2)^2}{n+5}$
Resolve the indeterminate form that arises when calculating the limit of the following difference of rational sequences:…
Resolve the indeterminate form that arises when calculating the limit of the following difference of rational sequences:
$\lim_{n \to \infty} \left( \frac{n^2 + 5n}{n + 2} - n \right)$
Sequences with Radicals
There are two situations. In a quotient with a root, divide by the highest power of $n$ and use $\sqrt{n^2}=n$ (valid because $n>0$). In a difference of roots ($\infty-\infty$), multiply and divide by the conjugate, just as you did to rationalize denominators.
If the difference is multiplied by $n$, as in $n\left(\sqrt{n^2+a}-n\right)$, the conjugate step still comes first. Then you simplify and compare degrees.
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 presents a double square root and requires applying the conjugate technique carefully:…
Calculate the limit of the following sequence, which presents a double square root and requires applying the conjugate technique carefully:
$\lim_{n \to \infty} n \cdot \left( \sqrt{n^2 + 4} - n \right)$
Working Backward: Finding Parameters
Sometimes the limit is given and you must find an unknown coefficient. The idea: the limit is only finite if the leading terms cancel perfectly, and that forces one parameter. The conjugate then gives the other.
The same steps work for any leading coefficient. Suppose the root is $\sqrt{c^2n^2+an+k}$ with $c>0$. Then the root behaves like $cn$, so you need $b=c$. The conjugate gives $\dfrac{an+k}{\sqrt{c^2n^2+an+k}+cn}\to\dfrac{a}{2c}$.
Find the values of the real parameters $a$ and $b$ (with $b > 0$) so that the following equality is satisfied:…
Find the values of the real parameters $a$ and $b$ (with $b > 0$) so that the following equality is satisfied:
$\lim_{n \to \infty} \left( \sqrt{9n^2 + an + 2} - bn \right) = 4$
Geometric and Exponential Sequences
A geometric sequence has the form $a_n=a\cdot r^n$. What happens depends only on the common ratio $r$.
Study the behavior and calculate the limit of the exponential sequence: $f_n = 12 \cdot \left(\frac{3}{4}\right)^n$
Study the behavior and calculate the limit of the exponential sequence:
$f_n = 12 \cdot \left(\frac{3}{4}\right)^n$
The temperature in degrees Fahrenheit 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…
The temperature in degrees Fahrenheit 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.
Geometric Series: Adding Infinitely Many Terms
Add the terms of a geometric sequence with ratio $r$: $a+ar+ar^2+\cdots$. The partial sum after $n$ terms is
$$S_n=\sum_{k=1}^{n}a\,r^{k-1}=a\cdot\frac{1-r^n}{1-r}\qquad(r\ne1).$$If $|r|<1$, then $r^n\to0$, so the partial sums converge.
Many problems add a fixed starting amount to a series. Then the total after $n$ steps looks like $L_n=c+\sum_{k=1}^{n}a\,r^{k-1}$, and its limit is $c+\dfrac{a}{1-r}$.
Figures can hide geometric series too. Take a square of side $6$ (area $36$), join the midpoints of its sides to make a new square, and repeat. The figure shows the first four squares.
Each new square has half the area of the one before it. The areas form a geometric sequence with $a=36$ and $r=\frac12$, so the total area is $\dfrac{36}{1-\frac12}=72$.
A high-tech elastic cable is subjected to iterative tension tests. Its initial length at rest is $20$ feet. In the first test it stretches $4$ feet. In each successive…
A high-tech elastic cable is subjected to iterative tension tests. Its initial length at rest is $20$ feet. In the first test it stretches $4$ feet. In each successive test, the material loses elasticity and stretches a $75\%$ of what it stretched in the immediately preceding test. The total length after $n$ tests is given by $L_n = 20 + \sum_{k=1}^n 4 \cdot (0.75)^{k-1}$. Calculate the theoretical maximum length that the cable will reach if subjected to infinite tests.
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…
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. Knowing 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 constructed in this way: $S_\infty = \lim_{n \to \infty} (A_1 + A_2 + \dots + A_n)$.
The figure shows the first three iterations in the construction of the Koch curve. If we start from an initial segment of length $L_0 = 1$ foot (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 $L_0 = 1$ foot (panel A), at each step $n$ the total length of the curve is multiplied by $\frac{4}{3}$ with respect 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?
Limits of Recurrence Sequences
A sequence can be defined by a rule such as $a_{n+1}=f(a_n)$. You may also see the same rule written with the previous term, $a_n=f(a_{n-1})$. It is the same idea with the index shifted by one. If you know the sequence converges to some $L$, then both $a_n$ and $a_{n+1}$ (or $a_{n-1}$) tend to $L$. So $L$ must satisfy the fixed-point equation $L=f(L)$.
The graph below shows the idea for $f(x)=\frac{x}{3}+4$. The limit is where the line $y=f(x)$ meets the line $y=x$.
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 $C_n = 15 + C_{n-1} \cdot 0.4$, where $15 \text{ mg/L}$ is the contribution of the new dose and the factor $0.4$ indicates that a $40\%$ of the drug from the previous day remains. If the initial concentration is $C_1 = 15$, calculate the equilibrium concentration in the blood in the long term, that is, $\lim_{n \to \infty} C_n$.
A sequence is defined by the following recurrence law: the first term is $a_1 = \sqrt{6}$, and each subsequent term is obtained as $a_{n+1} = \sqrt{6 + a_n}$. Assuming…
A sequence is defined by the following recurrence law: 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 (so it is convergent and has a limit $L$), set up the equation that the limit must satisfy and find it.
The Number e
Look at $\left(1+\dfrac{1}{n}\right)^n$. The base tends to $1$, and $1$ to any power is $1$, but the exponent tends to $\infty$, and a power that big can make a tiny excess grow. These two forces fight. The result is a new constant.
More generally, for any real number $k$:
$$\lim_{n\to\infty}\left(1+\frac{k}{n}\right)^n=e^k\qquad\text{and}\qquad\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{kn}=e^k.$$Why? Put $m=n/k$. Then $\left(1+\frac{k}{n}\right)^n=\left(1+\frac1m\right)^{km}=\left[\left(1+\frac1m\right)^m\right]^k\to e^k$.
This is where continuous compounding comes from. If $P$ dollars earn interest at annual rate $r$, compounded $n$ times a year, the balance after $t$ years is $P\left(1+\frac{r}{n}\right)^{nt}$. Letting $n\to\infty$ gives $Pe^{rt}$.
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:
$g_n = \left(1 + \frac{1}{n}\right)^{7n}$
$\$5000$ are deposited in an investment fund that offers an annual interest rate of $3.5\%$. If the financial institution compounds the interest $n…
$\$5000$ are deposited in an investment fund that offers an annual interest rate of $3.5\%$. If the financial institution compounds the interest $n$ times a year, the accumulated balance after $10$ years is given by the sequence $Cn = 5000 \cdot \left(1 + \frac{0.035}{n}\right)^{10n}$. Calculate the limiting balance that would be obtained if the compounding were continuous ($n \to \infty$).
The Form $1^\infty$ in General
Often the base is not written as $1+\frac1n$. For example, $\dfrac{n+5}{n+1}$ tends to $1$, but it hides the "$1+$" part. The next rule handles it.
Calculate the limit of the sequence, transforming the expression to apply the limit of the number $e$: $\lim_{n \to \infty} \left( \frac{n + 7}{n + 2} \right)^{4n}$
Calculate the limit of the sequence, transforming the expression to apply the limit of the number $e$:
$\lim_{n \to \infty} \left( \frac{n + 7}{n + 2} \right)^{4n}$
Resolve the following limit that gives rise to an indeterminate form of the type $1^\infty$, paying special attention to the algebraic operations in the exponent:…
Resolve the following limit that gives rise to an indeterminate form of the type $1^\infty$, paying special attention to the algebraic operations in the exponent:
$\lim_{n \to \infty} \left( \frac{n^2 - n + 3}{n^2 + n + 1} \right)^{n^2 - 1}$
Back to the start: the cocoa temperature $70+90\,(0.9)^n$ is a convergent geometric-type sequence. Its limit is $70$ °F, the room temperature, and the distance to the limit shrinks but never vanishes.