A limit is never just a calculation: when you say "the temperature of the coffee tends to $22\,°C$" or "the average cost tends to $5\,€$ per unit", you are using the limit to predict what happens in the very long run without having to wait for it to happen. In this section you will apply everything you have learned (limits, continuity, asymptotes) to problems set in real contexts.
Many phenomena (cooling, populations, the concentration of a drug) approach a stable value in the long run. That value is, precisely, the limit as the variable tends to infinity.
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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…
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…
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?
Tiered tariffs, progressive taxes and salaries with commission are often defined piecewise. Studying their continuity detects whether the model has any "unfair jumps".
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A phone company charges according to the function $C(x)$ (in euros), where $x$ is the number of gigabytes consumed in a month:…
A phone company charges according to the function $C(x)$ (in euros), where $x$ is the number of gigabytes consumed in a month:
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 salary.
The quotient $\dfrac{e(2+h)-e(2)}{h}$ (distance travelled between two very close instants) looks like a $\frac00$: the numerator and the denominator both tend to $0$ at the same time. If we simplify before letting $h\to0$, the limit reveals the instantaneous velocity. This is the first idea of the derivative, which you will meet in the next unit.
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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 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 $t = 2$.
Limits with the indeterminate form $\infty-\infty$ in context
When two expressions that grow without bound are subtracted, the result is not obvious: the expression must be transformed (conjugate, common factor) before taking the limit, just as with sequences.
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The capital accumulated in a bank account with continuous compound interest can be modelled by the limit…
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.
The long-term profit of a technology company is given by the difference between two revenue and cost forecasting models:…
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.
When a model mixes two different types of growth (exponential and polynomial, for example), Bolzano's theorem lets us prove that an equilibrium point exists without having to solve the equation.
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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…
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.
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:…
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.