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 $h(x) = \frac{2x+1}{x-4}$.

Study the continuity of the function $f(x) = \frac{1}{x+5} - 2$. What type of discontinuity does it have at its singular point?

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

Given the function $f(x) = \frac{3x-6}{x-2}$, calculate the limit as $x \to 2$. Take into account whether it is an indeterminate form or a vertical asymptote.

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

Determine the value of the following limits for the function $f(x) = \frac{1-x}{x}$:

1. a) $\lim_{x \to 0^-} f(x)$
2. b) $\lim_{x \to 0^+} f(x)$
3. c) $\lim_{x \to +\infty} f(x)$

To organize an excursion, the cost of renting the bus is $600$ dollars, which will be shared equally among the attendees. The function that defines 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 a 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 $4$ gallons:

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

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

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

It is known that the function $f(x) = \frac{ax+2}{3x-b}$ has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 4$.

Calculate the values of the parameters $a$ and $b$.

The intensity of an acoustic signal $I$ (in watts per square foot) received by a microphone decreases in inverse proportion to the distance $d$ (in feet) from the speaker, 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 speaker.
2. b) Calculate $\lim_{d \to +\infty} I(d)$ and interpret the result.

Find the value of the parameter $k$ so that the piecewise function is continuous at $x = 1$:

$f(x) = \begin{cases} \frac{k}{x+1} & \text{if } x < 1 \\ 2x - 3 & \text{if } x \ge 1 \end{cases}$

The temperature $T$ (in degrees Fahrenheit) of an electronic component as a function of the time $t$ (in minutes) since the cooling fan is turned on is given by $T(t) = 25 + \frac{80}{t+2}$.

1. a) What is the temperature at the instant the fan is turned on ($t=0$)?
2. b) Calculate the limit of $T(t)$ as $t \to +\infty$. What does this value represent in reality?

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 dollars) to remove $p$ percent of the pollutants from a lake is modeled by the function $C(p) = \frac{120p}{100-p}$, valid for $0 \le p < 100$.

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.

An electronic component manufacturing company has fixed monthly costs of $2000$ dollars and a variable cost of $15$ dollars for each unit produced. The average cost function per unit, if $x$ units are produced, is $C_m(x) = \frac{2000 + 15x}{x}$.

1. a) Study the continuity of $C_m(x)$ in the context of the problem ($x > 0$).
2. b) Calculate $\lim_{x \to +\infty} C_m(x)$ and prove analytically why the average cost per unit can never be exactly $15$ dollars, no matter how many units are manufactured.

In biochemistry, Michaelis-Menten kinetics describes the rate of 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 rate 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 rate of the reaction to be exactly half of its maximum asymptotic limit.

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

1. a) Find its inverse function $f^{-1}(x)$.
2. b) Calculate the horizontal and vertical asymptotes of both $f(x)$ and $f^{-1}(x)$.
3. c) Explain the geometric and analytical relationship that exists between the asymptotes of a homographic function and those of its inverse.

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

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

The piecewise function is defined as:

$f(x) = \begin{cases} \frac{x+k}{x-2} & \text{if } x < 1 \\ x^2 - a & \text{if } x \ge 1 \end{cases}$

Determine the values of the constants $a$ and $k$ knowing that $\lim_{x \to -\infty} f(x) = 1$ and that the function is continuous at $x = 1$.

Let the function be $f(x) = \frac{x+1}{x-1}$.

1. a) Determine the analytical expression of the composite function $(f \circ f)(x) = f(f(x))$, simplifying the result as much as possible.
2. b) In view of the result, what is the limit of $(f \circ f)(x)$ as $x \to 1$? Is there any contradiction with the domain of the original function? Reason your answer.

We want to construct a rectangle with a fixed area equal to $36 \text{ cm}^2$. If we call $x$ the length of the base, the perimeter 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) Interpret geometrically what shape the rectangle takes in each of these two limiting cases and why the perimeter tends to infinity in both situations.