Limit and Continuity Problems
Put a hot chocolate from the cafeteria on your desk. You are sure it will end up at room temperature, and you do not have to wait two hours to know it. A limit is the math behind that kind of prediction: it tells you where a quantity is heading without waiting for it to get there. In this section you will use limits, continuity and asymptotes on problems with real context.
Long-Run Behavior and Horizontal Asymptotes
Many things settle toward a steady value: a cooling drink, a population, the amount of medicine in your blood. That steady value is the limit as the variable tends to infinity. On a graph, it is a horizontal asymptote.
Why does the degree rule work? Divide the top and bottom by the highest power of $t$ in the denominator. Every lower-power term then becomes a fraction like $\frac{c}{t^n}$ that goes to $0$.
The temperature $T$ (in degrees Fahrenheit) of a cup of coffee as a function of time $t$ (in minutes) is given by the expression $T(t) = 22 + 65 e^{-0.05t}$. Calculate…
The temperature $T$ (in degrees Fahrenheit) of a cup of coffee as a function of time $t$ (in minutes) is given by the expression $T(t) = 22 + 65 e^{-0.05t}$. Calculate the limit of the temperature as time tends to infinity and interpret the physical meaning of this result.
The number of individuals in a deer population in a nature reserve is modeled by the function $P(t) = \frac{1500t + 400}{2t + 1}$, where $t$ is the time elapsed in years…
The number of individuals in a deer population in a nature reserve is modeled by the function $P(t) = \frac{1500t + 400}{2t + 1}$, where $t$ is the time elapsed in years since 2020. Towards what value does the population tend in the very long term?
The concentration of a drug in the blood (in mg/L) after $t$ hours of its administration is given by $C(t) = \frac{50t}{t^2 + 4}$. Calculate the limit of $C(t)$ as…
The concentration of a drug in the blood (in mg/L) after $t$ hours of its administration is given by $C(t) = \frac{50t}{t^2 + 4}$. Calculate the limit of $C(t)$ as $t \to \infty$ and reason what happens to the drug in the body over time.
Limiting Values: Average Cost, Logistic Growth, Geometry
The same idea applies to costs, to epidemics and even to geometry. In each case we ask where a quantity is heading.
Limits can also describe where a moving point lands. Here a line through the fixed point $A(0, 2)$ and a point $P$ on the parabola $y = x^2$ crosses the $x$-axis at $Q$.
The average cost of producing $x$ units of an electronic component is $C_m(x) = \frac{3000 + 5x}{x}$ dollars. Calculate the limit of $C_m(x)$ as $x \to \infty$ and…
The average cost of producing $x$ units of an electronic component is $C_m(x) = \frac{3000 + 5x}{x}$ dollars. Calculate the limit of $C_m(x)$ as $x \to \infty$ and explain its economic meaning in terms of production costs.
The spread of a virus in a city follows the logistic model $N(t) = \frac{100000}{1 + 4999 e^{-0.2t}}$, where $N(t)$ is the number of infected people and $t$ are the days…
The spread of a virus in a city follows the logistic model $N(t) = \frac{100000}{1 + 4999 e^{-0.2t}}$, where $N(t)$ is the number of infected people and $t$ are the days elapsed since the start of the outbreak. Calculate the number of infected people at the initial instant ($t=0$) and the limit as $t \to \infty$. What does this limit represent in the epidemiological context?
A point $P(x, x^2)$ moves along the parabola $y = x^2$ in the first quadrant. A straight line is drawn passing through the fixed point $A(0, 1)$ and the point $P$. This…
A point $P(x, x^2)$ moves along the parabola $y = x^2$ in the first quadrant. A straight line is drawn passing through the fixed point $A(0, 1)$ and the point $P$. This line intersects the x-axis at a point $Q$. Calculate the coordinates of point $Q$ in terms of $x$ and find the limit of the position of $Q$ as point $P$ moves toward infinity ($x \to \infty$).
Continuity of Piecewise Models
Phone plans, parking fees and pay scales are often defined in pieces. Checking continuity at the point where the pieces meet tells you whether the model has an unfair sudden jump.
A telephone company charges according to the function $C(x)$ (in dollars), where $x$ are the gigabytes consumed in a month:…
A telephone company charges according to the function $C(x)$ (in dollars), where $x$ are the gigabytes consumed in a month:
$C(x) = \begin{cases} 15 & \text{if } 0 \le x \le 10 \\ 15 + 2(x - 10) & \text{if } x > 10 \end{cases}$
Study the continuity of the function at $x = 10$ by calculating the one-sided limits and indicate if there is any sudden jump in the rate.
One-Sided Limits and Limits That Blow Up
Sometimes a model only makes sense on one side of a point. A tank has water only until $t = 60$ minutes, so we can only approach $60$ from the left. We write $t \to 60^-$. Other times the values do not settle at all: they grow without bound.
A water tank of $100$ gallons is emptied by opening a valve. The remaining volume of water follows the function $V(t) = 100\left(1 - \frac{t}{60}\right)^2$, where $t$ is…
A water tank of $100$ gallons is emptied by opening a valve. The remaining volume of water follows the function $V(t) = 100\left(1 - \frac{t}{60}\right)^2$, where $t$ is in minutes ($0 \le t \le 60$). Calculate the limit of $V(t)$ as $t \to 60^-$ and interpret the result.
In the theory of relativity, the kinetic energy of a particle with rest mass $m_0$ moving at speed $v$ is…
In the theory of relativity, the kinetic energy of a particle with rest mass $m_0$ moving at speed $v$ is $E_c = m_0 c^2 \left( \frac{1}{\sqrt{1 - v^2/c^2}} - 1 \right)$, where $c$ is the speed of light. Calculate the limit of $E_c$ as $v \to c^-$. What physical implication does this result have regarding the possibility of a massive particle reaching the speed of light?
Instantaneous Speed as a Limit
In Algebra 1 you found an average rate of change: distance divided by time over an interval. To get the speed at a single instant, shrink the interval to nothing. The quotient looks like $\frac{0}{0}$, so you must simplify first.
The table shows the average speed on $[3, 3+h]$ closing in on $11$:
| $h$ | 1 | 0.1 | 0.01 |
|---|---|---|---|
| average speed (ft/s) | 12 | 11.1 | 11.01 |
Drag the slider to watch the secant line turn into the tangent line at $t = 3$.
This limit is the first idea of a derivative, the main tool of calculus.
The distance traveled by a moving body in feet is given by the function $e(t) = t^2 + 3t$. The average speed between the instants $t = 2$ and $t = 2+h$ is calculated…
The distance traveled by a moving body in feet is given by the function $e(t) = t^2 + 3t$. The average speed between the instants $t = 2$ and $t = 2+h$ is calculated using the quotient $\frac{e(2+h) - e(2)}{h}$. Calculate the limit of this expression as $h \to 0$ to find the instantaneous speed at $t = 2$.
Holes, Jumps and Removable Discontinuities
If direct substitution gives $\frac{0}{0}$, the numerator and denominator share a factor. Factor, cancel, and then substitute. The limit exists even though the original formula is not defined at that point.
Not every $\frac{0}{0}$ is a hole. With an absolute value, split into the two sides.
| Type | What happens at $x = a$ | On the graph |
|---|---|---|
| Removable | the limit exists, but $f(a)$ is missing or different | open circle (hole) |
| Jump | the one-sided limits are different finite numbers | the graph jumps |
| Infinite | the values grow without bound | vertical asymptote |
When a demand formula such as $\dfrac{p^2 - 100}{p - 10}$ has a hole at $p = 10$, the cause is the way the formula is written: it is a fraction that cannot be evaluated at that price. The trend on both sides is clear, and the limit tells you the value the model "wants" to have there.
The graph shows the demand function $D(p) = \frac{p^2 - 100}{p - 10}$ for a product, where $p$ is the price in dollars and $D(p)$ is the quantity demanded, valid for…
The graph shows the demand function $D(p) = \frac{p^2 - 100}{p - 10}$ for a product, where $p$ is the price in dollars and $D(p)$ is the quantity demanded, valid for $p \neq 10$. Calculate analytically the limit of the demand as the price $p$ approaches $10$ dollars. Why do you think the mathematical model has a discontinuity (a "hole") exactly at $p=10$?
The efficiency of a solar panel as a function of the temperature $T$ (in degrees Fahrenheit) is given by the expression $E(T) = \frac{T^3 - 27}{T^2 - 9}$ for values…
The efficiency of a solar panel as a function of the temperature $T$ (in degrees Fahrenheit) is given by the expression $E(T) = \frac{T^3 - 27}{T^2 - 9}$ for values close to $T = 3$ (a critical temperature in a laboratory test). Calculate the limit of the efficiency as $T \to 3$ to determine the continuous theoretical value at that point.
The vertical trajectory of a drone in a flight test has been programmed according to the function $h(t) = \frac{|t^2 - 9|}{t - 3}$ for $t \neq 3$, where $h$ is the…
The vertical trajectory of a drone in a flight test has been programmed according to the function $h(t) = \frac{|t^2 - 9|}{t - 3}$ for $t \neq 3$, where $h$ is the height in feet and $t$ is the time in seconds. To avoid a software failure, the engineer must assign a value to $h(3)$. Study the one-sided limits at $t=3$. Is it possible to define $h(3)$ so that the function is continuous? Explain the physical consequences of this behavior.
Finding Parameters That Make a Piecewise Function Continuous
To avoid unfair jumps, a model's unknown constants are chosen so the pieces connect. At each join, set the left-hand limit equal to the right-hand limit.
With two unknowns, you need two joins, which give a system of two equations.
The monthly salary of a salesperson is $S(x) = \begin{cases} 1200 + 10x & \text{if } 0 \le x \le 50 \\ a + 15x & \text{if } x > 50 \end{cases}$, where $x$ is the number…
The monthly salary of a salesperson is $S(x) = \begin{cases} 1200 + 10x & \text{if } 0 \le x \le 50 \\ a + 15x & \text{if } x > 50 \end{cases}$, where $x$ is the number of sales made. Find the value of the parameter $a$ so that the function is continuous at $x = 50$, thus avoiding an unjustified jump in the salary.
The percentage of taxes $I(x)$ paid by a company as a function of its profits $x$ (in hundreds of thousands of dollars) is governed by the following law:…
The percentage of taxes $I(x)$ paid by a company as a function of its profits $x$ (in hundreds of thousands of dollars) is governed by the following law:
$I(x) = \begin{cases} \frac{x^2 - 9}{x - 3} & \text{if } 0 \le x < 3 \\ ax + b & \text{if } 3 \le x \le 6 \\ 12 & \text{if } x > 6 \end{cases}$
Calculate the values of the parameters $a$ and $b$ so that the tax function is continuous throughout its domain.
The concentration of a chemical pollutant in a river is modeled by the function:…
The concentration of a chemical pollutant in a river is modeled by the function:
$C(t) = \begin{cases} \frac{t^2 - 16}{t - 4} & \text{if } 0 \le t < 4 \\ at + b & \text{if } 4 \le t \le 8 \\ \frac{2t^2 - 32}{t - 4} & \text{if } t > 8 \end{cases}$
Find the values of $a$ and $b$ so that the concentration model is a continuous function at all times.
Unknown Parameters in Rational Functions
Sometimes the facts about a model are given as limits and values, and you must find the coefficients. Turn each fact into an equation.
An epidemic propagation model estimates that the number of infected people (in thousands) follows the rational function $I(t) = \frac{at^2 + bt + c}{t^2 + 1}$, where $t$…
An epidemic propagation model estimates that the number of infected people (in thousands) follows the rational function $I(t) = \frac{at^2 + bt + c}{t^2 + 1}$, where $t$ is the time in weeks. It is known that in the very long term the number of infected people stabilizes at $50$ thousand, that at the initial instant $t=0$ there were $10$ thousand infected people, and that the limit of the function as $t \to 1$ is $35$. Find the parameters $a, b, c$.
The demand function for a luxury item is $D(p) = \frac{p^2 + ap + b}{p^2 - 4p + 3}$, where $p$ is the price in thousands of dollars. It is known that the function has a…
The demand function for a luxury item is $D(p) = \frac{p^2 + ap + b}{p^2 - 4p + 3}$, where $p$ is the price in thousands of dollars. It is known that the function has a vertical asymptote at $p = 1$ but does NOT have a vertical asymptote at $p = 3$ (the discontinuity at $p=3$ is removable). Knowing that the limit of the demand as $p \to 3$ is $2$, find the parameters $a$ and $b$.
The Indeterminate Form $\infty - \infty$
When two quantities that grow without bound are subtracted, the result is not obvious. It might be $0$, a number or $\infty$. You must rewrite the expression before taking the limit. With square roots, multiply by the conjugate. With fractions, combine them over a common denominator, or divide first to pull out the part that grows.
When each fraction has a top degree greater than its bottom degree, a common denominator is messy. Instead, divide each numerator by its denominator (long division or synthetic division). Each fraction becomes a linear part plus a leftover fraction whose limit is $0$. Then the linear parts can be compared directly.
The long-term profit of a technology company is given by the difference between two revenue and cost forecast models: $B(t) = \sqrt{t^2 + 50t + 100} - \sqrt{t^2 + 10t}$,…
The long-term profit of a technology company is given by the difference between two revenue and cost forecast models: $B(t) = \sqrt{t^2 + 50t + 100} - \sqrt{t^2 + 10t}$, where $t$ is the time in months. Calculate the limit of the profit as $t \to \infty$ by multiplying and dividing by the conjugate expression.
The long-term maintenance cost of two computer servers is given by the expressions $C_1(x) = \frac{x^3 + 2x^2}{x^2 - 1}$ and $C_2(x) = \frac{x^2 + 5x}{x + 1}$, where $x$…
The long-term maintenance cost of two computer servers is given by the expressions $C_1(x) = \frac{x^3 + 2x^2}{x^2 - 1}$ and $C_2(x) = \frac{x^2 + 5x}{x + 1}$, where $x$ is the number of months of continuous use. We want to know if the difference in cost between the two servers stabilizes over time. Calculate $\lim_{x \to \infty} (C_1(x) - C_2(x))$.
The Number $e$ and the Form $1^\infty$
A bank can compound interest $n$ times a year: $A = P\left(1 + \dfrac{r}{n}\right)^{nt}$. What happens if $n$ grows without bound (compounding continuously)? The base tends to $1$ and the exponent tends to $\infty$, so the result is not obvious.
| $n$ | 1 | 10 | 100 | 1,000 | 10,000 |
|---|---|---|---|---|---|
| $\left(1 + \frac{1}{n}\right)^n$ | 2 | 2.5937 | 2.7048 | 2.7169 | 2.7181 |
To get the interest formula, let $m = \dfrac{n}{r}$, so $n = mr$. Then $\left(1 + \dfrac{r}{n}\right)^{nt} = \left(1 + \dfrac{1}{m}\right)^{mrt} = \left[\left(1 + \dfrac{1}{m}\right)^m\right]^{rt}$. As $n \to \infty$, $m \to \infty$, and the bracket tends to $e$. So the balance is $A = Pe^{rt}$.
The same trick works for any limit of the form $1^\infty$.
The capital accumulated in a bank account with continuous compound interest can be modeled by the limit…
The capital accumulated in a bank account with continuous compound interest can be modeled by the limit $C(t) = \lim_{n \to \infty} C_0 \left(1 + \frac{r}{n}\right)^{nt}$, where $n$ is the number of compounding periods per year. If the initial capital is $\$C0 = 5000$ and the annual rate is $r = 0.04$, calculate the limit to obtain the continuous compounding formula and determine the balance after $t = 10$ years.
In a bacteria culture under controlled conditions, the biomass after $t$ hours is modeled by the function…
In a bacteria culture under controlled conditions, the biomass after $t$ hours is modeled by the function $B(t) = 50 \left( \frac{t^2 + 3t + 1}{t^2 + t + 5} \right)^{2t}$. Knowing that this is an indeterminate form of the type $1^\infty$, calculate the limit of the biomass as $t \to \infty$.
Oscillations: $\dfrac{\sin x}{x}$ and the Squeeze Idea
Damped oscillations, such as a pendulum slowing down, mix an oscillating factor with a decaying one. Two tools handle them.
For a damped pendulum $A(t) = e^{-kt}\dfrac{\sin(ct)}{t}$, the limit as $t \to 0^+$ gives the starting amplitude (the formula is undefined at $t = 0$, but the values approach a number). The limit as $t \to \infty$ gives the long-term behavior.
A damped pendulum is released from an initial position. Its amplitude of oscillation is given by the function $A(t) = e^{-0.1t} \frac{\sin(2t)}{t}$ for $t > 0$.…
A damped pendulum is released from an initial position. Its amplitude of oscillation is given by the function $A(t) = e^{-0.1t} \frac{\sin(2t)}{t}$ for $t > 0$. Calculate the limit of the amplitude as $t \to 0^+$ (remember that $\lim_{x \to 0} \frac{\sin x}{x} = 1$) and as $t \to \infty$. Interpret the behavior of the pendulum both at the initial instant and in the long term.
The Intermediate Value Theorem
When a model mixes different kinds of growth, such as exponential and linear, you cannot always solve $f(x) = 0$ exactly. The Intermediate Value Theorem still proves a solution exists.
The idea: a continuous graph has no breaks, so to get from below the $x$-axis to above it, it has to cross.
A company estimates that its monthly revenue (in thousands of dollars) follows the function $I(x) = 2^x - x^2$, and its operating costs are $C(x) = \frac{10}{x+1}$,…
A company estimates that its monthly revenue (in thousands of dollars) follows the function $I(x) = 2^x - x^2$, and its operating costs are $C(x) = \frac{10}{x+1}$, where $x$ is the price of the product in dollars ($x \ge 0$). Prove, justifying your answer using Bolzano's Theorem, that there is a price $x$ in the interval $[4, 5]$ for which the company has a strictly zero profit.
Summary
Back to the start: the temperature of a hot drink has the form (room temperature) $+\, c\,e^{-kt}$. The exponential term tends to $0$, so the limit is the room temperature. That is the prediction you made without waiting.