Exercises: Asymptotes of a function

Observe the following graph representing the RAM memory usage (in GB) of a process server as a function of time $t$ (in seconds). Intuit the values of the one-sided limits at time $t = 4$ and reason whether the overall limit exists at that point.

A technical inspection drone follows an approach path to an antenna whose height is given by the function $H(x) = \frac{x^2 - 9}{x - 3}$, where $x$ is the horizontal distance to the control centre. Calculate the limit of the height when the drone is very close to $x = 3$ metres.

(Note: Although the function is not defined at $x=3$, study the limit by simplifying the expression).

Calculate the following direct limits by substitution, representing the velocity of a particle in metres per second at different times $t$:

1. a) $\lim_{t \to 2} (t^2 + 3t - 1)$

1. b) $\lim_{t \to 0} \frac{e^t + 1}{t + 2}$

Given the rational function $f(x) = \frac{5}{x - 2}$, which describes the intensity of a magnetic field relative to the distance $x$ to a conducting wire:

1. a) Determine the domain of the function.
2. b) Calculate the one-sided limits $\lim_{x \to 2^-} f(x)$ and $\lim_{x \to 2^+} f(x)$.
3. c) State the equation of its vertical asymptote.

In a biological study, the number of individuals in a bacterial population (in thousands) follows the model $P(t) = \frac{4t + 10}{t + 2}$, where $t$ is the time in hours.

1. a) Calculate $\lim_{t \to +\infty} P(t)$ to find the long-term population.
2. b) What type of asymptote does this result represent? Write its equation.

Find all asymptotes (vertical and horizontal) of the function $f(x) = \frac{3x - 6}{x + 1}$. To do so:

1. Identify the value of $x$ that makes the denominator zero.

2. Calculate the limit of the function as $x \to \infty$.

Determine the behaviour at infinity of the following polynomial functions, stating whether they tend to $+\infty$ or $-\infty$:

1. a) $f(x) = -2x^3 + 5x - 1$ as $x \to +\infty$

1. b) $g(x) = x^4 - 100x^2$ as $x \to -\infty$

Given the function $f(x) = \frac{x^2}{x^2 + 4}$, which models the efficiency of a photovoltaic panel as a function of the radiation received $x$, justify by calculating limits whether the function has any horizontal asymptote, and sketch it graphically.

In a Mathematics II exam, students are asked to analyse the asymptotes of the function $f(x) = \dfrac{x^2 - x - 2}{x^2 - 4}$. Lucas provides the following reasoning:

"Since the denominator vanishes at $x = 2$ and at $x = -2$, the function necessarily has two vertical asymptotes: the lines $x = 2$ and $x = -2$. Moreover, since $\lim_{x \to \infty} f(x) = 1$, the line $y = 1$ is a horizontal asymptote and, therefore, the graph of the function can never cross said line $y = 1$".

1. a) Identify and explain the two conceptual mistakes made by Lucas in his reasoning.
2. b) Correct the exercise: calculate the relevant one-sided limits at $x = -2$ to characterise the true vertical asymptote, explain what happens exactly at $x = 2$, and algebraically check whether or not the curve crosses the line $y = 1$.

Consider the rational function $f(x) = \dfrac{3x^2 - 5x}{x^2 + 1}$.

1. a) Estimate: Before applying any formal limit rule, look at the highest-degree terms of the numerator and the denominator. Reasonably predict what numerical value the function will tend to as $x$ takes very large positive values (for instance, $x = 1000$). Also deduce intuitively whether the curve will approach this value from above or from below, analysing the effect of the term $-5x$.
2. b) Check: Formally calculate the equation of the horizontal asymptote using the limits as $x \to +\infty$ and as $x \to -\infty$. Then, analytically study the relative position of the graph with respect to the asymptote by calculating the sign of the difference $f(x) - y_{\text{asymptote}}$. Assess whether the results confirm or correct your initial estimate.

Given the rational function $f(x) = \dfrac{ax^2 + 5x - 3}{2x^2 - 8}$, determine the value of the real parameter $a$ knowing that the function has a horizontal asymptote on the line $y = 3$. Once the value of $a$ has been found, calculate the one-sided limits of the function at its vertical asymptotes and explain the behaviour of the graph at those points.

A team of engineers is monitoring the temperature $T$ (in degrees Celsius) of a new graphics processor as a function of time $t$ (in minutes) since it was switched on. The function is given by:

Calculate the initial temperature of the processor and determine, by evaluating limits, whether the temperature stabilises in the long term. What type of asymptote does this stabilised value represent?

Observe the following graph of a function $f(x)$ (each tick mark on the axes corresponds to one unit). Assume that the two branches continue outside the box with the trend they show. From the visual information, determine the value of the following limits and the equations of all its asymptotes:

1. a) $\displaystyle \lim_{x \to -\infty} f(x)$

1. b) $\displaystyle \lim_{x \to 1^-} f(x)$ and $\displaystyle \lim_{x \to 1^+} f(x)$

1. c) $\displaystyle \lim_{x \to +\infty} f(x)$

Find the equation of the oblique asymptote of the function $f(x) = \dfrac{2x^2 - 3x + 1}{x - 2}$. To complete the study, state the relative position of the curve with respect to this asymptote as $x \to +\infty$ and as $x \to -\infty$.

Calculate the value of the following limit, which leads to an indeterminate form of type $1^\infty$, using its relation to the number $e$:

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

1. a) Calculate $\displaystyle \lim_{x \to +\infty} f(x)$.
2. b) Check whether a horizontal asymptote exists as $x$ tends to $+\infty$.
3. c) Investigate what happens to the limit of the function as $x \to -\infty$. Does any asymptote exist on that branch?

Given the function $f(x) = \dfrac{x^2 - k}{x^2 - 4x + 4}$:

1. a) Determine whether there is any value of $k$ for which the function has a removable discontinuity at $x = 2$.
2. b) If $k = 9$, study the existence of vertical asymptotes and describe the behaviour of the function nearby using one-sided limits.

Calculate the following limits by resolving the corresponding indeterminate forms:

1. a) $\displaystyle \lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{x}$

1. b) $\displaystyle \lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}$

To find the oblique asymptote of the rational function $f(x) = \dfrac{2x^3 - 3x^2 + 5}{x^2 - 1}$ and study the position of the curve with respect to it, two classmates propose different strategies:

• Strategy A (Algebraic division): Perform the Euclidean division of the numerator by the denominator to express the function as $f(x) = mx + n + \dfrac{R(x)}{Q(x)}$.

• Strategy B (Analytical limits): Calculate the coefficients using the formulae $m = \lim_{x \to \infty} \dfrac{f(x)}{x}$ and $n = \lim_{x \to \infty} [f(x) - mx]$.

1. a) Apply Strategy A: obtain the quotient and the simplified remainder, and state directly the equation of the asymptote line and the relative position of the curve at $+\infty$ and at $-\infty$.
2. b) Apply Strategy B: calculate both limits in detail until arriving at the same equation.
3. c) Compare both approaches: justify which of the two strategies is more efficient in this particular case to determine the relative position of the curve and explain in what type of functions Strategy A becomes completely unusable, making it essential to resort to Strategy B.

Design the analytical expression of a rational function $f(x) = \dfrac{P(x)}{Q(x)}$ that simultaneously satisfies the following four conditions:

1. It has a single vertical asymptote at the line $x = 3$, satisfying $\lim_{x \to 3^-} f(x) = -\infty$ and $\lim_{x \to 3^+} f(x) = +\infty$.

2. It has a removable discontinuity at the point with abscissa $x = -1$.

3. It has an oblique asymptote with equation $y = 2x - 1$.

4. It intersects the y-axis at the point $(0, -2)$.

1. a) Deduce, with reasoning, what the minimum degrees of the polynomials $P(x)$ and $Q(x)$ must be.
2. b) Construct the polynomials $P(x)$ and $Q(x)$ step by step, justifying how you guarantee each of the specified conditions.

A new data compression algorithm for a streaming platform is being designed. The system response time, in milliseconds, is modelled by the function $T(x)$, where $x$ represents the workload in terabytes:

Given that, for the system to be stable under extreme loads, the limit of the response time as the load tends to infinity must be exactly $e^4$, determine with reasoning the value of the parameter $a$.

Consider the function $f(x)$, defined by means of an absolute value, which models the trajectory of a surveillance drone:

Study the existence of oblique asymptotes of this function. If they exist, calculate their equations and analyze the position of the curve relative to those asymptotes both at $+\infty$ and at $-\infty$.

A communications satellite follows an orbit whose gravitational perturbation is described by the function $g(x)$. The satellite must not stray indefinitely from a predicted linear trajectory. The function is:

Find the values of the real parameters $a$ and $b$ so that the function has a horizontal asymptote at $y = 2$ as $x \to +\infty$.

Given the function dependent on the parameters $k$ and $m$:

Calculate the values of $k$ and $m$ so that $f(x)$ has an oblique asymptote passing through the point $(0, 2)$ with a gradient of $3$. Once found, check whether the function has any vertical asymptote and justify whether it is removable or not.

In the study of the spread of a computer virus in a social network, it has been determined that the number of infected nodes $N(t)$ after a time $t$ (in hours) follows the function:

Analyse the long-term behaviour of the infection by calculating $\lim_{t \to +\infty} N(t)$. Does the function have any horizontal asymptote? Justify your answer by applying the corresponding hierarchies of infinities.

Find the domain and all asymptotes of the following logarithmic function:

Determine the one-sided limits at each point where the function is undefined, and sketch the behaviour of the function near its vertical asymptotes.

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

Calculate the value of the following limit using L'Hôpital's Rule after performing the necessary transformations to resolve the indeterminate form:

From the result obtained, state whether the function $f(x) = \frac{sen(x) - x}{x^2 \cdot sen(x)}$ has a vertical asymptote at the origin.

Let $f(x)$ be the piecewise function defined as:

1. Find the value of $k$ so that the function has a finite jump discontinuity at $x = 2$.

2. For the value of $k$ obtained in the previous section, determine the value of $a$ so that the function has a horizontal asymptote $y = 3$ as $x \to +\infty$.

In the context of the energy transition in Spain, the average cost of electricity generation of two facilities is modelled as a function of the monthly energy produced $x$ (in thousands of MWh, with $x > 0$):

• A solar thermal plant with storage in Extremadura has an average unit cost (in euros/MWh) modelled by: $$C_1(x) = \dfrac{35x + \sqrt{16x^2 + 20x + 9}}{x}$$

• An experimental floating offshore wind farm has an average unit cost modelled by: $$C_2(x) = \dfrac{2x^2 + 15x + 8}{x + 1}$$

1. a) Reasonably study the behaviour of $C_1(x)$ for minimum production levels ($x \to 0^+$) and determine its horizontal asymptote when production grows on a large scale ($x \to +\infty$). Interpret the physical and management meaning of both asymptotes.
2. b) Analytically determine the asymptote of $C_2(x)$ as $x \to +\infty$ (horizontal or oblique), obtaining its complete equation.
3. c) Reasonably explain which of the two models describes a technology with sustainable long-term returns to scale and which one will present increasing unit costs when expanding production, relying on the typology of their asymptotes.

Consider the family of functions defined for all real numbers by the expression:

where $a$ and $b$ are real parameters.

1. a) Reasonably determine the values of $a$ and $b$ so that the graph of $f(x)$ has a horizontal asymptote at $+\infty$ whose equation is $y = 4$.
2. b) With the values of $a$ and $b$ set in the previous section, prove that the function does not have a horizontal asymptote at $-\infty$, but an oblique asymptote. Rigorously calculate its equation $y = mx + n$, paying special attention to the algebraic treatment of the radical when $x \to -\infty$ when factoring out terms.
3. c) Justify why this function cannot have any vertical asymptotes in the entire set of real numbers $\mathbb{R}$.