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.

To organize a trip, the cost of renting the bus is $12\,000$ pesos, 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 $C(x) = \frac{12\,000}{x}$.

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 $4$ liters:

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 meter) received by a microphone decreases inversely proportional to the distance $d$ (in meters) 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 $x \to 2$.

The cost $C$ (in thousands of pesos) to remove $p$ percent of the pollutants from a lake is modeled by the function $C(p) = \frac{2\,400p}{100-p}$, valid for $0 \le p < 100$.

1. a) Calculate the cost of removing $50\%$ and $90\%$ of the pollutants.
2. 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 $v(S) = \frac{V_{max} \times S}{K_m + S}$, 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.