Exercises: Inverse proportion function

Given the inverse proportion function $f(x) = \frac{4}{x-3}$, determine its domain and calculate the one-sided limits as $x$ tends to $3$.

Calculate the limits of the function $g(x) = \frac{-5}{x}$ as $x \to +\infty$ and as $x \to -\infty$. Interpret the result geometrically.

Find the equation of the horizontal asymptote and the vertical asymptote of the homographic function $h(x) = \frac{2x+1}{x-4}$.

Given that the function $f(x) = \frac{k}{x}$ passes through the point $(2, 6)$, find the value of the constant $k$ and calculate $\lim_{x \to 0^+} f(x)$.

The time $t$ (in hours) taken to travel a distance of $120$ km as a function of the average speed $v$ (in km/h) is given by $t(v) = \frac{120}{v}$. Calculate the limit of $t(v)$ as $v \to +\infty$ and explain its physical meaning.

Marcos has to find all the asymptotes of the function $f(x) = \frac{6}{2x - 8}$. On his exam, he writes the following solution:

"The function is of the linear fractional type $\frac{ax+b}{cx+d}$. For the vertical asymptote, I set the denominator equal to zero: $2x - 8 = 0 \implies x = 4$, so the line is $x = 4$. For the horizontal asymptote, I divide the first numbers that appear in the fraction, that is, the numerator by the coefficient of $x$: $y = \frac{6}{2} = 3$".

1. a) Identify the conceptual error Marcos made when determining the horizontal asymptote and explain what value the coefficient $a$ of $x$ in the numerator of this function actually has.
2. b) Justify analytically the correct horizontal asymptote by computing $\lim_{x \to +\infty} f(x)$ and explain the graphical meaning of this result.

Consider the rational function $f(x) = \frac{2x + 1}{x - 3}$.

1. a) Without calculating limits formally, estimate the approximate value of $f(x)$ for $x = 1000$ by comparing the relative weight of the linear terms versus the constant terms in the numerator and denominator. Using this estimate, conjecture what the horizontal asymptote line should be.
2. b) Estimate the sign of the result when you approach $3$ from the left ($x = 2,99$) and from the right ($x = 3,01$). Toward what infinite value do you intuit the function tends in each case?
3. c) Rigorously check your estimates by computing the limits $\lim_{x \to +\infty} f(x)$, $\lim_{x \to 3^-} f(x)$, and $\lim_{x \to 3^+} f(x)$, precisely deducing the equations of its two asymptotes.

To organise a trip, the cost of renting the bus is $600$ euros, which will be split equally among those attending. The function defining the cost per person as a function of the number of attendees $x$ is $C(x) = \frac{600}{x}$.

Calculate $\lim_{x \to 0^+} C(x)$ and $\lim_{x \to +\infty} C(x)$. Interpret both results in the context of the problem.

Boyle's Law states that, at constant temperature, the volume $V$ occupied by a gas is inversely proportional to the pressure $P$ to which it is subjected, that is, $V(P) = \frac{k}{P}$. If at a pressure of $1,5$ atmospheres the gas occupies a volume of $4$ litres:

1. a) Find the value of the constant $k$.
2. b) Calculate $\lim_{P \to +\infty} V(P)$ and explain what happens to the gas if the pressure increases indefinitely.

Given the function $f(x) = \frac{3x-1}{x+2}$:

1. a) Calculate the intersection points with the coordinate axes.
2. b) Find the equations of its asymptotes by evaluating the corresponding limits.

The intensity of an acoustic signal $I$ (in watts per square metre) received by a microphone decreases inversely proportional to the distance $d$ (in metres) from the loudspeaker, following the model $I(d) = \frac{150}{d}$.

1. a) Calculate $\lim_{d \to 0^+} I(d)$ and explain why the microphone cannot be placed right next to the loudspeaker.
2. b) Calculate $\lim_{d \to +\infty} I(d)$ and interpret the result.

Determine the domain and study the continuity of the function $f(x) = \frac{x^2-4}{x-2}$. Is it a pure inverse proportion function? Justify your answer by calculating the limit as $x \to 2$.

A youth association from Soria is organizing a cultural trip whose coach has a fixed rental cost of €$480$, regardless of the number of passengers. In addition, the agency charges a fixed fee of €$5$ per attendee for individual travel insurance.

1. a) Write the function $C(x)$ representing the average total cost each person must pay if $x$ people attend the trip.
2. b) Express the function $C(x)$ in the canonical form of a linear fractional function $C(x) = \frac{ax+b}{cx+d}$ and state the admissible domain of the function in this context if the coach has a maximum of $55$ seats.
3. c) Compute $\lim_{x \to +\infty} C(x)$, determine the equation of the horizontal asymptote, and interpret its practical meaning: what price would the ticket per person tend to if the coach had unlimited capacity? Why can the cost per person never be equal to or less than €$5$?

You want to find the algebraic expression of a linear fractional function of the form $f(x) = \frac{ax + b}{x + d}$ that simultaneously satisfies three conditions:

1. Its graph has the line $x = -2$ as a vertical asymptote.

2. Its graph has the line $y = 3$ as a horizontal asymptote.

3. It intersects the y-axis ($Y\text{-axis}$) at the point $(0, -1)$.

1. a) State the appropriate mathematical conditions and calculate the numerical values of the constants $a$, $b$, and $d$.
2. b) Verify that the values obtained satisfy the non-degeneracy condition $ad - bc \ne 0$ (considering $c = 1$) and explain what geometric defect the graph would have if it had turned out that $ad - bc = 0$.
3. c) Using the resulting function, compute the one-sided limits $\lim_{x \to -2^-} f(x)$ and $\lim_{x \to -2^+} f(x)$ to describe the position of the branches of the hyperbola relative to the vertical asymptote.

The cost $C$ (in thousands of euros) of removing $p$ percent of the pollutants from a lake is modeled by the function $C(p) = \frac{120p}{100-p}$, valid for $0 \le p < 100$. The figure shows the graph of this function.

1. a) Calculate the cost of removing $50\%$ and $90\%$ of the pollutants.

1. b) Calculate $\lim_{p \to 100^-} C(p)$ and explain, from an economic and environmental point of view, why it is impossible to remove $100\%$ of the pollutants.

In biochemistry, Michaelis-Menten kinetics describes the rate of an enzymatic reaction $v$ as a function of the substrate concentration $S$. The equation is $v(S) = \frac{V_{max} \cdot S}{K_m + S}$, where $V_{max} = 100 \,\mu\text{mol/s}$ is the maximum velocity and $K_m = 5 \text{ mM}$ is a constant.

1. a) Calculate $\lim_{S \to +\infty} v(S)$ and state what $V_{max}$ represents graphically.
2. b) Find the substrate concentration $S$ required for the reaction rate to be exactly half of its maximum asymptotic limit.

Starting from the basic inverse proportion function $f(x) = \frac{1}{x}$, a translation of $3$ units to the right and $2$ units upwards is applied to obtain a new function $g(x)$.

1. a) Write the algebraic expression for $g(x)$ in the form $\frac{ax+b}{cx+d}$.
2. b) Prove, by calculating limits, that the asymptotes of $g(x)$ coincide with the translations applied to the coordinate axes.

We want to construct a rectangle of fixed area equal to $36 \text{ cm}^2$. If we let $x$ be the length of the base, the perimetre of the rectangle as a function of $x$ is given by $P(x) = 2x + \frac{72}{x}$.

1. a) Calculate $\lim_{x \to 0^+} P(x)$ and $\lim_{x \to +\infty} P(x)$.
2. b) Geometrically interpret what shape the rectangle takes in each of these two limiting cases and why the perimetre tends to infinity in both situations.

To study in detail the asymptotes and the position of the curve relative to them for the function $f(x) = \frac{4x - 5}{x - 2}$, two classmates debate which method is more convenient:

• Method 1 (Analytical limits): Find the horizontal asymptote using $\lim_{x \to \pm\infty} f(x)$, the vertical one using $\lim_{x \to 2^\pm} f(x)$, and to determine whether the curve is above or below the horizontal asymptote, solve for the sign of the difference $f(x) - y_{\text{asymptote}}$.

• Method 2 (Polynomial division and translation): Divide the numerator by the denominator to write the function as $f(x) = C + \frac{R}{x - 2}$ (that is, a translated basic hyperbola of the type $\frac{k}{x}$), deducing in one step the asymptotes $y = C$, $x = 2$, and the position of the branches depending on the sign of the remainder $R$.

1. a) Fully carry out Method 1: find the asymptotes and analytically determine the position of the curve relative to the horizontal asymptote at $+\infty$ and at $-\infty$.
2. b) Fully carry out Method 2: perform polynomial division, express $f(x)$ as the sum of an integer quotient plus a proper fraction, and explain how the position of the branches is deduced from this expression without needing to set up additional inequalities.
3. c) Argue critically: which of the two methods do you consider more intuitive and faster for sketching the hyperbola by hand and identifying its center of symmetry? And which is more useful and generalizable for analyzing asymptotes in rational functions where the degree of the numerator exceeds that of the denominator?

The cost $C(p)$ (in hundreds of thousands of euros) to clean up and remove $p$ percent ($0 \le p < 100$) of the accumulated plastic waste in a section of the Ebro river basin is modeled by the function:

where $k$ is a positive constant related to the technology used.

1. a) Analyze the one-sided limit $\lim_{p \to 100^-} C(p)$ and interpret the result in the context of the problem: why does the line $p = 100$ act as a vertical asymptote, and what real economic implication does aiming for the absolute decontamination ($100\,\%$) of that section have?
2. b) Knowing that cleaning $50\,\%$ of the plastics costs €$120\,000$ ($1,2$ hundred thousand euros), calculate the exact value of the constant $k$.
3. c) With the value of $k$ obtained, a spokesperson states: "If we double the investment allocated to $50\,\%$ (reaching €$240\,000$), we will manage to clean $100\,\%$ of the waste". Mathematically refute this statement by calculating the actual percentage $p$ of waste that would be cleaned with that doubled budget.
4. d) Calculate the marginal cost of going from cleaning $40\,\%$ to $50\,\%$, and compare it to going from $80\,\%$ to $90\,\%$. Explain why, despite both cases involving an identical increase of $10\,\%$, the difference in costs is so disproportionate, connecting your reasoning to the asymptotic behavior of the function.