If you share the cost of hiring a coach among more and more friends, each person pays less: with $2$ people, $300\,€$ each; with $100$, barely $6\,€$. That relationship —the more people, the lower the cost per person, but never $0\,€$— is inverse proportionality, and its graph is a curve with two branches that never touch the axes: the hyperbola.
Given that the function $f(x) = \frac{k}{x}$ passes through the point $(2, 6)$, find the value of the constant $k$ and calculate $\lim_{x \to 0^+} f(x)$.
Given that the function $f(x) = \frac{k}{x}$ passes through the point $(2, 6)$, find the value of the constant $k$ and calculate $\lim_{x \to 0^+} f(x)$.
When $x$ grows without bound, $\dfrac{k}{x}$ becomes as small as you like: it gets close to $0$ but never reaches it. The line $y=0$ is its horizontal asymptote.
Near $x=0$ the opposite happens: the quotient shoots off towards $+\infty$ or $-\infty$ depending on the sign of $k$ and the side from which you approach. The line $x=0$ is its vertical asymptote.
Practice★☆☆Not started
Calculate the limits of the function $g(x) = \frac{-5}{x}$ as $x \to +\infty$ and as $x \to -\infty$. Interpret the result geometrically.
Calculate the limits of the function $g(x) = \frac{-5}{x}$ as $x \to +\infty$ and as $x \to -\infty$. Interpret the result geometrically.
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…
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 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…
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.
Adding a constant to the numerator or the denominator, or shifting the $x$, translates the basic hyperbola without changing its shape. The result is a homographic function, whose asymptotes are no longer necessarily the axes.
Practice★☆☆Not started
Find the equation of the horizontal asymptote and the vertical asymptote of the homographic function $h(x) = \frac{2x+1}{x-4}$.
Find the equation of the horizontal asymptote and the vertical asymptote of the homographic function $h(x) = \frac{2x+1}{x-4}$.
Given the function $f(x) = \frac{3x-1}{x+2}$: a) Calculate the intersection points with the coordinate axes. b) Find the equations of its asymptotes by evaluating the…
Given the function $f(x) = \frac{3x-1}{x+2}$:
a) Calculate the intersection points with the coordinate axes.
b) Find the equations of its asymptotes by evaluating the corresponding limits.
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…
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 $g(x)$.
a) Write the algebraic expression for $g(x)$ in the form $\frac{ax+b}{cx+d}$.
b) Prove, by calculating limits, that the asymptotes of $g(x)$ coincide with the translations applied to the coordinate axes.
Not every fraction with $x$ in the denominator is a "pure" inverse proportionality: if the numerator shares a root with the denominator, the fraction simplifies and a removable discontinuity appears instead of an asymptote.
Practice★★☆Not started
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…
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 $x \to 2$.
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 $x$…
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 $P(x) = 2x + \frac{72}{x}$.
a) Calculate $\lim_{x \to 0^+} P(x)$ and $\lim_{x \to +\infty} P(x)$.
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.
Back to the beginning: The cost per person of the coach is $C(x) = \dfrac{600}{x}\,€$; by calculating $\displaystyle\lim_{x\to+\infty}\frac{600}{x}=0$ we see that the price tends to $0$, but since $y=0$ is a horizontal asymptote that the curve never touches, the trip will never be completely free.
Summary
Let's practice! 7 medium exercises in this section