Exercises: Problems on limits and continuity

The temperature $T$ (in degrees Celsius) 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 natural park is modelled by the function $P(t) = \frac{1500t + 400}{2t + 1}$, where $t$ is the time elapsed in years since 2020. What value does the population tend towards in the long term?

A phone company charges according to the function $C(x)$ (in euros), where $x$ is the number of 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 state whether there is a sudden jump in the tariff.

The distance travelled by a moving body in metres is given by the function $e(t) = t^2 + 3t$. The average speed between times $t = 2$ 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 $t = 2$.

The average cost of producing $x$ units of an electronic component is $C_m(x) = \frac{3000 + 5x}{x}$ euros. Calculate the limit of $C_m(x)$ as $x \to \infty$ and explain its economic meaning in terms of production costs.

The concentration of a drug in the blood (in mg/L) $t$ hours after its administration is given by $C(t) = \frac{50t}{t^2 + 4}$. Calculate the limit of $C(t)$ as $t \to \infty$ and explain what happens to the drug in the body over time.

A $100$-litre water tank 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.

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 salary.

To analyze the long-term cost difference between two industrial processes, Sofía needs to calculate the following limit:

On her exam, she writes the following:

"Since for very large values of $x$ the term $6x$ is negligible compared to $x^2$, we have $\sqrt{x^2 + 6x} \approx \sqrt{x^2} = x$. Therefore, the limit is $\lim_{x \to +\infty} (x - x) = 0$. The long-term difference will be zero."

1. a) Explain with reasons why Sofía's argument is mathematically incorrect and what type of indeterminate form she has overlooked.
2. b) Solve the limit correctly using the conjugate technique and calculate the true equilibrium value to which the cost difference approaches.

A municipal water company in Spain bills bi-monthly domestic consumption through a tiered regulated tariff to encourage savings. If a household consumes $x$ cubic metres ($m^3$), the total bill amount $C(x)$ (in euros) is governed by the function:

where $k$ is a constant set by the city council.

1. a) Interpret the practical meaning of the values $12$ and $0,80$ in the first tier of the model.
2. b) Determine the value that the parameter $k$ must have so that the billing function is continuous at $x = 15\text{ m}^3$.
3. c) Justify what fairness issue or economic inconsistency for the user would occur if a value of $k$ different from the one obtained were chosen (distinguish between $k$ being greater or smaller).

The percentage tax rate $I(x)$ paid by a company as a function of its profits $x$ (in hundreds of thousands of euros) is governed by the following rule:

$I(x) = \begin{cases} \frac{x^2 - 9}{x - 3} & \text{if } 0 \le x < 3 \\ ax + b & \text{if } 3 \le x \le 6 \\ 12 & \text{if } x > 6 \end{cases}$

Calculate the values of the parameters $a$ and $b$ so that the tax function is continuous throughout its domain.

The 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 $t$ is the time in days since the start of the outbreak. Calculate the number of infected people at the initial time ($t=0$) and the limit as $t \to \infty$. What does this limit represent in the epidemiological context?

The graph shows the demand function $D(p) = \frac{p^2 - 100}{p - 10}$ for a product, where $p$ is the price in euros and $D(p)$ is the quantity demanded, valid for $p \neq 10$. Calculate analytically the limit of the demand as the price $p$ approaches $10$ euros. Why do you think the mathematical model presents a discontinuity (a "hole") at exactly $p=10$?

The efficiency of a solar panel as a function of the temperature $T$ (in degrees Celsius) 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 theoretical continuous value at that point.

The capital accumulated in a bank account with continuous compound interest can be modelled 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 $C_0 = 5000$ euros and the annual rate is $r = 0,04$, calculate the limit to obtain the formula for continuous capital and determine the balance after $t = 10$ years.

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?

The concentration of a chemical pollutant in a river is modelled 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.

The long-term profit of a technology company is given by the difference between two revenue and cost forecasting 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.

During a time-trial stage of the Vuelta a España, the distance covered by a cyclist along a straight section is given by the expression $e(t) = 0,5t^2 + 4t$, where $t$ is measured in seconds and $e(t)$ in metres (valid for $0 \le t \le 30$). You want to find their exact instantaneous velocity at time $t = 8\text{ s}$.

1. a) Calculate the average velocity over the time intervals $[8;\, 9]$, $[8;\, 8,1]$, and $[8;\, 8,01]$. From these numerical values, make an intuitive estimate of what the exact instantaneous velocity at $8\text{ s}$ should be.
2. b) Check your estimate by algebraically calculating the limit of the difference quotient defining this velocity:

Expand the numerator, simplify the fraction before substituting $h$ with $0$, obtain the exact value, and interpret the units of the result.

We want to prove that the interior temperature of an automated greenhouse in Almería equals the exterior temperature at some point between $2$ and $3$ in the morning. The thermal balance between both environments leads to the equation $x^3 - 2x^2 - 1 = 0$, where $x$ is the time in hours elapsed since midnight.

To justify that there is at least one solution in the interval $[2, 3]$, two members of the team propose different approaches:

• Method A (Direct algebraic): Try to solve the equation exactly by factoring the polynomial using Ruffini's rule with the possible integer roots.

• Method B (Topological/Continuity): Define a continuous function $f(x) = x^3 - 2x^2 - 1$ and apply Bolzano's Theorem on the interval $[2, 3]$.

1. a) Apply Method A by checking whether any of the possible integer roots of the polynomial allows solving the equation on said interval. What limitation do you find?
2. b) Apply Method B by checking each of the hypotheses of Bolzano's Theorem in detail.
3. c) Compare both methods and justify which one is appropriate for this problem, concluding rigorously whether or not the stated thermal equalization occurs.

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 $Q$ as a function of $x$ and find the limit of the position of $Q$ as point $P$ moves away towards infinity ($x \to \infty$).

You can do it on paper and send a photo with your answer.

An epidemic spread 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 long term the number of infected people stabilises at $50$ thousand, that at the initial time $t=0$ 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 vertical trajectory of a drone during 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 metres and $t$ is the time in seconds. To avoid a software glitch, 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 behaviour.

The long-term maintenance cost of two IT 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 whether the cost difference between both servers stabilises over time. Calculate $\lim_{x \to \infty} (C_1(x) - C_2(x))$.

A company estimates that its monthly revenue (in thousands of euros) 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 euros ($x \ge 0$). Show, justifying your answer using Bolzano's Theorem, that there is a price $x$ in the interval $[4, 5]$ for which the company makes exactly zero profit.

In a bacterial culture under controlled conditions, the biomass after $t$ hours is modelled by the function $B(t) = 50 \left( \frac{t^2 + 3t + 1}{t^2 + t + 5} \right)^{2t}$. Given that this is an indeterminate form of the type $1^\infty$, calculate the limit of the biomass as $t \to \infty$.

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 euros. 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). Given that the limit of the demand as $p \to 3$ is $2$, find the parameters $a$ and $b$.

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 behaviour of the pendulum both at the initial instant and in the long term.

A pharmacological bioengineering team in Valencia is designing the administration protocol for a drug. They need to construct a continuous function $C(t)$ that models the drug concentration in blood plasma (in $\text{mg/l}$) as a function of time $t \ge 0$ (in hours), strictly meeting the following design conditions set by the physicians:

1. During the first two hours ($0 \le t \le 2$), the concentration must grow quadratically according to the law $C(t) = k \cdot t^2$, starting from $0$ and reaching exactly $6\text{ mg/l}$ at time $t = 2$.

2. For $t > 2$, the drug stabilizes and is eliminated slowly according to a rational model of the form $C(t) = \dfrac{a \cdot t + b}{t - 1}$.

3. The concentration cannot undergo abrupt jumps at any time for physiological safety reasons (the function must be continuous at $t = 2$).

4. In the very long term ($t \to +\infty$), if the patient receives basal maintenance microdoses, the residual equilibrium concentration must converge exactly to $1,5\text{ mg/l}$.

1. a) Deduce with justification the value of the constant $k$ in the first interval.
2. b) Using the long-term asymptotic behavior and the continuity requirement, set up the corresponding system and uniquely find the parameters $a$ and $b$.
3. c) Write the final analytical expression of the function $C(t)$ and formally verify that all required conditions are satisfied (one-sided continuity at $t = 2$ and limit at infinity).

In an industrial area, the concentration of a gaseous pollutant in the air (in $\mu\text{g/m}^3$) at a distance $x$ (in km) downwind from the emission source is given by the model:

Regional air quality regulations mandate that the maximum permissible level for that area is given by the reference threshold $U(x) = \dfrac{25}{x + 3}$.

1. a) Calculate the limit as $x \to +\infty$ of the theoretical function $C(x)$ (algebraically resolving the indeterminate form using the conjugate) and interpret what this asymptotic value would physically mean if the model were applicable at infinite distances.
2. b) Environmental technicians suspect that, even though $C(0) = 0$ and $U(0) > 0$, there is necessarily a critical point in the interval $[1, 6]$ where the actual pollutant level exactly equals the regulatory threshold. Prove this statement rigorously by formulating the appropriate function and verifying all the hypotheses of Bolzano's Theorem, without needing to calculate the exact position.
3. c) From $x = 12\text{ km}$ onwards, a vegetation barrier is implemented that alters dispersion according to the model $C_{\text{forest}}(x) = \dfrac{A \cdot x}{x + 3}$. Determine with reasons the exact value of the parameter $A$ so that the pollutant concentration does not have a jump discontinuity at the junction point $x = 12$.

Unit challenge. How much does each kilowatt-hour consumed in your home really cost? Model the unit cost to optimize household energy spending.

1. Data collection: Check the latest electricity bill for your home (or consult the CNMC/REE simulator). Note down the monthly fixed cost ($F$, approx. $15,20$ €) and the variable price ($p$, approx. $0,14$ €/kWh). State the rational function for average cost per kWh: $C(x) = \frac{F + p \cdot x}{x}$.

2. Tables and analytical calculation: In GeoGebra or a spreadsheet, create a table to numerically estimate $\lim_{x \to 0^+} C(x)$. Analytically find the intercepts, sign, intervals of monotonicity, and asymptotes.

3. Continuity and limits: Algebraically calculate $\lim_{x \to +\infty} C(x)$ by resolving the indeterminate form. Analyze the continuity of the function on $[0, +\infty)$ and explain what both asymptotes mean in real life.

4. Reasoned proposal: Write a brief report with the resulting graph and two practical conclusions on the impact of phantom power consumption and energy savings.

Assessment criteria:

• Precision in the analytical calculation of limits, asymptotes, sign, and continuity analysis.

• Correct use of GeoGebra or a spreadsheet for table estimation and plotting.

• Rigour in the economic interpretation of asymptotic behaviour based on real data.