Exercises: Asymptotes and infinite branches

Given the rational function $f(x) = \frac{3x - 6}{x + 2}$, carry out the following study:

1. a) Identify its vertical asymptote.

1. b) Calculate the limit of the function as $x$ tends to $+\infty$ to find its horizontal asymptote.

The download speed of a file is being monitored in Megabytes per second (MB/s). The function describing the speed after $t$ seconds is:

$v(t) = \frac{10t}{t + 5}$

Find the infinite branches by calculating the limit as $t \to +\infty$. What value does the download speed stabilise at in the long term?

Analyze and sketch the function $f(x) = \frac{1}{x^2 + 1}$ by following these steps:

1. a) Find the domain.

1. b) Check whether it has any symmetry.

1. c) Determine whether there are any horizontal asymptotes.

1. d) Using the information above, make a rough drawing of the graph.

You can do it on paper and send a photo with your answer.

Consider a function $f(x)$ that satisfies the following three characteristics:

1. It has a vertical asymptote at $x = 0$.

2. It is an always increasing function on its domain.

3. It crosses the $X$ axis at the point $(1, 0)$.

Propose a simple analytical expression (formula) that satisfies these three conditions simultaneously and check that each one of them holds.

The figure shows the graph of a rational function and its two asymptotes.

• a) Read the domain of the function from the graph.

• b) Write the equations of the vertical asymptote and the horizontal asymptote.

• c) State the position of the curve relative to the horizontal asymptote in each branch.

Calculate the vertical and horizontal asymptotes of the function $f(x) = \frac{2x + 1}{x - 4}$, writing down the limits that justify them.

Consider the function $f(x) = e^x + 2$.

• a) Does it have vertical asymptotes? Why or why not?

• b) Calculate $\lim_{x \to -\infty} f(x)$ and deduce its horizontal asymptote.

• c) Is the graph above or below this asymptote? And what happens when $x \to +\infty$?

True or false? Reason your answer with an example: "Every rational function has at least one vertical asymptote."

For the rational function $f(x) = \dfrac{x^2 - 4}{x^2 - 5x + 6}$, Marcos states on his Mathematics exam: "Since the values that make the denominator zero are $x = 2$ and $x = 3$, the function has two vertical asymptotes: the lines $x = 2$ and $x = 3$".\na) Identify the conceptual error made by Marcos and explain why his statement is false.\nb) Determine the true vertical and horizontal asymptotes of the function, calculating the necessary one-sided limits to justify the behavior of the graph at the problematic points.

A report by the Banco de España analyzes the implementation of the Bizum payment system among people over 65 years old in an autonomous community. The evolution of the percentage of users in this group is modeled by the function:\n$$P(t) = 85 \cdot (1 - e^{-0,3t})$$\nwhere $t \ge 0$ is the time elapsed in years since the start of the digitalization campaigns and $P(t)$ represents the penetration percentage over the total group.\na) Check whether the function has any vertical asymptote in its domain.\nb) Calculate the horizontal asymptote of the function, justifying the calculation of the limit as $t \to +\infty$.\nc) Interpret the socioeconomic meaning of the value obtained for the asymptote: according to this model, will a $100\,\%$ adoption rate ever be reached in this group? And $85\,\%$? Justify your answer.

Given the rational function $f(x) = \frac{ax+b}{x-3}$, determine the values of the real parameters $a$ and $b$ given that:

• The function has a horizontal asymptote at $y = 2$.

• The graph of the function passes through the point $P(1, -1)$.

Once $a$ and $b$ are found, study the monotonicity of the function and check whether it has any stationary points.

A sociological study on the spread of a rumour in a student community follows the function $P(t) = 100(1 - e^{-0,2t})$, where $P$ is the percentage of people who know the rumour and $t$ is the time in hours ($t \ge 0$).

1. Calculate the initial percentage of people who know the rumour.

2. What value does the percentage of people approach as time passes? Interpret the result in terms of asymptotes.

3. Justify, using the first derivative, that the rumour always spreads (the function is increasing).

4. Does the curvature of the function change at any point? Reason your answer.

Study the infinite branches and the existence of asymptotes of the function $f(x) = \frac{x^2 - 1}{x+2}$.

1. Find the equation of the oblique asymptote.

2. Determine the position of the curve relative to this asymptote as $x \to +\infty$ and as $x \to -\infty$.

3. Find the points on the graph where the tangent line is parallel to the oblique asymptote.

4. Draw an approximate sketch of the asymptotes and the function.

You can do it on paper and send a photo with your answer.

The diagram shows the graph of the function $f(x) = \frac{x^2 + 1}{x + 2}$ and its oblique asymptote.

• a) Find the equation of the oblique asymptote analytically by calculating $m = \lim_{x \to +\infty} \frac{f(x)}{x}$ and $n = \lim_{x \to +\infty} (f(x) - mx)$.

• b) Perform algebraic long division of the numerator by the denominator and check that you obtain the same result.

• c) Study the position of the curve relative to the oblique asymptote as $x \to +\infty$ and as $x \to -\infty$.

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

• a) Does it have vertical asymptotes? Justify your answer.

• b) Does it have a horizontal asymptote? Calculate it if it exists.

• c) Find its oblique asymptote and study the position of the curve relative to it.

A function $f$ satisfies $\lim_{x \to 1^-} f(x) = -\infty$, $\lim_{x \to 1^+} f(x) = +\infty$ and $\lim_{x \to +\infty} f(x) = 2$, as shown in the figure with two possible graphs.

• a) Which of the two graphs, Panel A or Panel B, could correspond to $f$? Justify your choice by checking each of the three limits.

• b) Write the equations of the asymptotes of $f$.

• c) Propose a simple formula for a function that satisfies these three conditions.

Study the infinite branches of the function $f(x) = \frac{3x^2 - 2}{x + 1}$: vertical asymptote, oblique asymptote, and position of the curve relative to the latter as $+\infty$ and as $-\infty$.

The average production cost of an independent magazine is $C(x) = \frac{500 + 2x}{x}$ euros per copy, where $x$ is the number of copies printed.

• a) Find the horizontal asymptote of $C(x)$ and interpret it: why can the average cost not fall below a certain value?

• b) What happens to the average cost when very few copies are printed? Relate this to the vertical asymptote.

• c) Does it make sense to speak of an oblique asymptote for this function? Justify your answer.

Design the simplified algebraic expression of a rational function of the form $f(x) = \dfrac{a x^2 + b x + c}{x^2 + d x + e}$ that simultaneously satisfies the following four conditions:\n1. It has a vertical asymptote at the line $x = 3$.\n2. It has a removable discontinuity at $x = -2$ (that is, the denominator vanishes at $x = -2$, but the function does not have a vertical asymptote at that point).\n3. It has a horizontal asymptote at the line $y = 2$.\n4. Its graph intersects the y-axis at the point $(0, 4)$.\n\nClearly explain the reasoning followed to deduce the factors of the numerator and denominator, and verify by formally calculating limits that your function satisfies the conditions for the vertical and horizontal asymptotes.

Consider the rational function $f(x) = \dfrac{2x^2 + 5x - 1}{x + 2}$.\nTo find its slant asymptote $y = m x + n$, two students propose different approaches:\n- Method 1 (General limits): Calculate $m = \displaystyle\lim_{x \to +\infty} \dfrac{f(x)}{x}$ and then $n = \displaystyle\lim_{x \to +\infty} \big(f(x) - m x\big)$.\n- Method 2 (Polynomial long division): Divide the numerator polynomial by the denominator polynomial to express the function as $\text{quotient} + \dfrac{\text{remainder}}{\text{divisor}}$.\n\na) Apply both methods detailing all the steps and verify that the same line is obtained.\nb) Justify which of the two methods is more efficient for rational functions and explain what immediate additional information the remainder of the division provides regarding the relative position of the curve with respect to the asymptote as $x \to +\infty$ and as $x \to -\infty$.

Let $f(x) = \dfrac{ax^2 + b}{x - 2}$ be a rational function. Determine the values of the real parameters $a$ and $b$, knowing that the function has an oblique asymptote that passes through the point $(0, 2)$ and that it has a relative extremum at the point with abscissa $x = 0$. Once you have found $a$ and $b$, carry out a complete study of the function (domain, intercepts with the axes, asymptotes, intervals of increase and decrease, and extrema) and graph it.

Analyze and graph the function $f(x) = \dfrac{e^x}{x^2 - 3x}$. Your study must include:

1. Domain and points of discontinuity.

2. Infinite limits and vertical asymptotes.

3. Horizontal asymptotes and parabolic branches (studying the behavior both at $+\infty$ and at $-\infty$).

4. Calculation of the first derivative to find intervals of monotonicity and relative extrema.

5. Sketch of the graph indicating all the elements you found (do it by hand in your notebook or with the tool you prefer).

Consider the function $f(x) = \frac{x^2 - 4}{x - 1}$.

• a) Determine its domain and its vertical asymptote, studying the one-sided limits.

• b) Find the oblique asymptote and study the position of the curve relative to it as $+\infty$ and as $-\infty$.

• c) Calculate the axes intercepts and sketch the complete graph with its asymptotes.

You can do it on paper and send a photo with your answer.

The diagram shows the graph of a function with a vertical asymptote and an oblique asymptote, as shown in the figure.

• a) From the graph, deduce its domain and the equations of its asymptotes.

• b) Propose a rational function whose graph could be that in the figure, and check that its asymptotes match those you deduced.

Consider the function $f(x) = x \cdot e^{1/x}$ for $x > 0$.

• a) Study its behaviour as $x \to 0^+$.

• b) Show that it has an oblique asymptote as $x \to +\infty$ and find its equation (hint: calculate $m = \lim_{x \to +\infty} \frac{f(x)}{x}$ first and recall that $\lim_{u \to 0} \frac{e^u - 1}{u} = 1$).

Consider the function $f(x) = \frac{\ln x}{x}$.

• a) Study the infinite branch as $x \to 0^+$. What type is it?

• b) Study the infinite branch as $x \to +\infty$, knowing that $\lim_{x \to +\infty} \frac{\ln x}{x} = 0$. What asymptote appears?

• c) At what point does the graph intersect this asymptote? State with reasons whether it is contradictory for a curve to intersect its own horizontal asymptote.

Determine the value of $b$ so that the function $f(x) = \frac{x^2 + a}{x + b}$ has a vertical asymptote at $x = 3$ and an oblique asymptote with equation $y = x + 3$.

• a) State the conditions that must be satisfied and find $b$.

• b) What condition must $a$ satisfy for the vertical asymptote to genuinely exist (that is, for the discontinuity to be non-removable)?

• c) Choose a valid value for $a$ and verify both asymptotes.

Compare the behaviour as x approaches infinity of $f(x) = \frac{x^2}{e^x}$ and $g(x) = \frac{e^x}{x^2}$.

• a) Study the infinite branches of $f$ as $+\infty$ and as $-\infty$, stating the type of each branch (you may use the fact that an exponential grows faster than any power).

• b) Do the same for $g$.

• c) In light of the results, explain why it is said that "exponential growth always dominates power growth".

A logistics platform located in Zaragoza (PLAZA) studies the average cost per processed package $\overline{C}(x)$ (in euros) as a function of the daily volume of shipments $x$ (measured in hundreds of packages, with $x > 0$), according to the model:\n$$\overline{C}(x) = \dfrac{0,2x^2 + 4x + 80}{x}$$\n\na) Estimate: Before calculating any asymptote, answer these two intuitive questions with justification:\n - If the daily activity is practically zero ($x \to 0^+$), is it logical for the unit cost to tend to stabilize or to skyrocket toward infinity? Relate it to the burden of fixed costs.\n - If the daily volume grows indefinitely ($x \to +\infty$), will the average cost per package tend to zero, to a fixed horizontal ceiling, or will it grow linearly due to logistics saturation issues?\nb) Calculate: Formally determine all infinite branches of the function (vertical, horizontal, and slant asymptotes) for $x > 0$, calculating the necessary limits.\nc) Check and evaluate: Compare the analytical results with your previous estimates. From the equation of the obtained asymptote, interpret what each of its terms represents economically when the shipment volume is very high.

An economic consultancy defines the viability index $V(x)$ of an investment project as a function of the market risk index $x \in \mathbb{R}$ using the following piecewise-defined function:\n$$V(x) = \begin{cases} \dfrac{a x^2 + 4x - 1}{x - 3} & \text{if } x < 2 \\[2ex] \dfrac{b x + 5}{x - 1} & \text{if } x \ge 2 \end{cases}$$\n\na) An analyst hastily claims: "The function has two vertical asymptotes: the lines $x = 3$ and $x = 1$, because they are the values that make the denominators zero". Justify rigorously why the analyst makes a serious conceptual error by ignoring the domains of definition of each branch.\nb) Determine the values of the real parameters $a$ and $b$ knowing that:\n - When the risk grows without bound ($x \to +\infty$), the viability index tends to stabilize at the value $3$.\n - At the opposite extreme ($x \to -\infty$), the function has a slant asymptote whose slope is $m = -2$.\nc) For the values of $a$ and $b$ found, calculate the full equation of this slant asymptote ($y = mx + n$) and study whether the function has any discontinuity at the junction point $x = 2$, classifying the type of discontinuity if any.