Exercises: Limit and continuity problems

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 since 2020. Towards what value does the population tend in the very long term?

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.

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 $t = 2$.

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 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.

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.

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:

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

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 $t$ 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?

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 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 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 $\$C_0 = 5000$ 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 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 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.

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.

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 ($x \to \infty$).

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 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.

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 $\lim_{x \to \infty} (C_1(x) - C_2(x))$.

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.

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 $t \to \infty$.

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 $b$.

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.