Exercises: Asymptotes and infinite branches in rational functions

Calculate the following limits by analysing the leading term of each polynomial:

1. a) $\lim_{x \to +\infty} (2x^3 - 5x^2 + 10)$

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

1. c) $\lim_{x \to +\infty} (4 - x^4)$

Given the following graph of the function $f(x)$, determine the value of the limits at infinity and state the equation of its horizontal asymptote:

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

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

Calculate the limits as $x \to +\infty$ and as $x \to -\infty$ of the following rational functions by comparing the degrees of the numerator and the denominator:

1. a) $f(x) = \frac{3x^2 - 5}{x^3 + 2x}$

1. b) $g(x) = \frac{x^2 + 1}{4x^2 - 3x}$

1. c) $h(x) = \frac{5x^3 + 2}{x^2 - 7}$

Find the equation of the horizontal asymptote of the following functions. If it does not have one, state so with justification:

1. a) $f(x) = \frac{2x - 1}{x + 3}$

1. b) $g(x) = \frac{4}{x^2 - 1}$

1. c) $h(x) = \frac{x^2 + 5}{2x - 3}$

Calculate the following limits of functions involving radicals:

1. a) $\lim_{x \to +\infty} \sqrt{x^2 - 3x}$

1. b) $\lim_{x \to +\infty} \frac{\sqrt{4x^2 + 1}}{x - 5}$

1. c) $\lim_{x \to +\infty} \frac{x+2}{\sqrt{x^3 + 1}}$

Match each of the following functions with its behaviour at $+\infty$:

1. $f(x) = \frac{10}{x-2}$

2. $g(x) = \frac{5x^2}{x^2 + 1}$

3. $h(x) = \frac{x^2 - 1}{x}$

Behaviours:

1. a) Tends to $+\infty$ (Parabolic branch).
2. b) Has a horizontal asymptote at $y = 0$.
3. c) Has a horizontal asymptote at $y = 5$.

For the function $f(x) = \dfrac{1}{x^2}$, complete the following table of values to investigate its behaviour as $x$ approaches $0$.

From the table, deduce the value of the one-sided limits $\lim_{x \to 0^-} f(x)$ and $\lim_{x \to 0^+} f(x)$. Does the overall limit exist at $x=0$?

Given the function $f(x) = \dfrac{x + 1}{x - 4}$:

1. a) State its domain of definition.
2. b) Calculate the one-sided limits at the point where the function is not defined: $\lim_{x \to 4^-} f(x)$ and $\lim_{x \to 4^+} f(x)$.
3. c) What type of infinite branch does the function have at $x = 4$?

Sofía is studying the vertical asymptotes of the rational function $f(x) = \frac{x^2 - 4x + 3}{x^2 - 1}$. To do this, she sets up the equation $x^2 - 1 = 0$, obtains the solutions $x = 1$ and $x = -1$, and states on her exam: "The function has two vertical asymptotes, whose equations are $x = 1$ and $x = -1$, because they are the two real values that make the denominator zero".

1. a) Explain in detail what conceptual error Sofía made in drawing that conclusion without first analyzing the numerator.
2. b) Correct her reasoning by calculating the necessary limits to determine what the true vertical asymptotes of $f(x)$ are and what exactly happens on the graph at the point that Sofía interpreted incorrectly.

Consider the function $g(x) = \frac{6x^2 - 5}{2x^2 + 3}$, which describes the time evolution of a financial indicator as a function of time $x$ (with $x > 0$).

1. a) Estimate: Without applying formal limit rules, observe only the highest-degree terms of the numerator and denominator for very large values of $x$ and predict the numerical value at which the indicator stabilizes in the long run. Intuitively reason whether the graph will approach that value from above or from below by mentally comparing the numerator with the denominator multiplied by your estimate.
2. b) Check: Rigorously calculate $\lim_{x \to +\infty} g(x)$ to confirm the equation of the horizontal asymptote and determine the relative position of the curve with respect to that line by studying the sign of the difference $g(x) - y_{\text{asymptote}}$. Compare your mathematical conclusion with the estimate you formulated.

Given the graph of the function $f(x)$ shown below, identify the equations of its asymptotes (vertical and horizontal) and describe the behaviour of the function using limits as $x \to +\infty$ and as $x \to -\infty$.

Given the rational function $f(x) = \frac{2x^2 - 8}{x^2 - x - 2}$:

1. Determine its domain of definition.

2. Calculate the one-sided limits at the points where the function is not defined.

3. Specify which of those points correspond to a vertical asymptote and which to a removable discontinuity (missing point).

Determine the value of the parameters $a$ and $b$ so that the function $g(x) = \frac{ax + 5}{bx - 4}$ has a vertical asymptote at the line $x = 2$ and a horizontal asymptote at the line $y = 3$. Once found, check whether the function is increasing or decreasing on its domain.

Study the behavior at infinity of the function $f(x) = \frac{x^2 - 3x}{x + 1}$ by calculating:

1. a) The existence of an oblique asymptote of the form $y = mx + n$.

1. b) The relative position of the curve with respect to that asymptote when $x \to +\infty$ (indicate whether the function lies above or below the line).

The number of users of an educational social network, measured in thousands, follows the time evolution $t$ (in months) according to the function:

1. Calculate the initial number of users (at $t=0$).

2. Determine what the number of users will tend to in the long term (as $t \to +\infty$).

3. Is there any value of $t$ for which the function is not continuous? Reason whether this value makes sense in the context of the problem.

Calculate the following limits and, from them, determine the horizontal asymptotes of each function:

1. $\lim_{x \to +\infty} \left( \sqrt{x^2 + 5x} - x \right)$

2. $\lim_{x \to -\infty} \frac{3^x + 2}{3^x - 1}$

Consider the function $f(x) = \frac{k}{x^2 - 9}$.

1. If we know that the function passes through the point $(0, -1)$, find the value of $k$.

2. With the value of $k$ obtained, carry out a complete study of the asymptotes of the function and position its infinite branches relative to its vertical asymptotes.

Study the infinite branches of the function $f(x) = \dfrac{x^3}{x^2 + 1}$. Having no horizontal or vertical asymptotes, determine the equation of its oblique asymptote and schematically represent its behaviour at infinity using the following reference graph:

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

1. Its graph has a horizontal asymptote at the line $y = -3$.

2. It has a single vertical asymptote at the line $x = 2$.

3. It passes through the origin of coordinates $(0, 0)$.

4. It has no removable discontinuities (that is, $P(x)$ and $Q(x)$ share no roots in the set of real numbers).

Once the function is proposed:

1. a) Justify step by step how you deduced the polynomials $P(x)$ and $Q(x)$ and verify that the conditions are met by calculating the relevant limits.
2. b) Reasonably explain why this function cannot have an oblique asymptote in either direction of infinity ($+\infty$ or $-\infty$).

The municipal bicycle rental service of a Spanish provincial capital records its operating costs. An economic report models the average cost per subscribed user per month, $C(x)$ (expressed in euros), as a function of the total number of active subscribers, $x$ (with $x \ge 1$), using the rational function:

1. a) If the customer acquisition campaign is highly successful and the number of subscribers grows indefinitely ($x \to +\infty$), what value does the monthly average cost per user approach? Mathematically interpret this value as a horizontal asymptote and explain its meaning for the economic viability of the municipal service.
2. b) Determine whether the function approaches its horizontal asymptote from above or from below as $x \to +\infty$. What economic implication does this behavior have for the city council as the subscriber base increases?
3. c) The algebraic expression has a vertical asymptote at $x = -50$. Justify why this asymptote completely lacks meaning in the context of the real problem.

Given the rational function $f(x) = \frac{ax^2 + bx + 1}{2x - 4}$, calculate the values of the parameters $a$ and $b$ knowing that the function has an oblique asymptote whose equation is $y = 2x + 5$. Once the parameters are found, study whether the graph of the function lies above or below this asymptote as $x \to +\infty$.

A digital marketing analyst is studying the evolution of the followers of a new social media account. The number of followers (in thousands) follows the function $S(t) = \frac{120t^2 + 500}{3t^2 + 4t + 10}$, where $t \geq 0$ is the time in months since the launch.

1. a) Determine whether there is a growth limit for the number of followers in the long term. If so, give the equation of the horizontal asymptote.

1. b) Graphically represent the behavior of the function at infinity by means of a sketch of its asymptotic branches (draw some axes, the asymptote and the curve approaching it when $t \to +\infty$).

Calculate all the asymptotes of the following piecewise-defined function and study the behaviour of its infinite branches:

$f(x) = \begin{cases} \frac{2x^2}{x+1} & \text{if } x < 0 \\ \frac{x-1}{x^2-9} & \text{if } x \geq 0 \end{cases}$

State specifically the one-sided limits at the points where vertical asymptotes exist.

Consider the family of functions $f(x) = \frac{x^2 - k}{x^2 + k^2 - 2}$. Find the values of the parameter $k \in \mathbb{R}$ for which the function:

1. Has no vertical asymptotes.

2. Has exactly one vertical asymptote.

3. Has two vertical asymptotes.

For all cases, state the horizontal asymptote of the function.

Given the function $f(x) = \sqrt{x^2 + 4x} - x$, whose domain is $(-\infty, -4] \cup [0, +\infty)$, study its behavior as $x \to +\infty$ and as $x \to -\infty$.

1. a) Calculate $\lim_{x \to +\infty} f(x)$ and, if it exists, the equation of the corresponding asymptote.

1. b) Calculate $\lim_{x \to -\infty} f(x)$ and, if there is an asymptote, find its equation (it may be horizontal or oblique).

1. c) Does it have the same horizontal asymptote in both directions? Justify your answer by calculating both limits in detail and draw a sketch of the resulting branches, indicating the asymptotes and the points $(-4, 4)$ and $(0, 0)$.

Let the function be $f(x) = \frac{|x^2 - 1|}{x - 1}$.

1. a) Redefine the function as a piecewise function by eliminating the modulus.

1. b) Calculate the one-sided limits at $x = 1$. Is there a vertical asymptote at that point or is it a removable discontinuity?

1. c) Study the existence of horizontal asymptotes by calculating $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$. Explain the difference between both results.

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

1. Has a vertical asymptote at $x = -4$.

2. Has a removable discontinuity at $x = 1$.

3. Has a horizontal asymptote at $y = 5$.

4. Passes through the origin $(0,0)$.

Provide the final algebraic expression of the function in simplified form.

Find the equation of the oblique asymptote of the function $f(x) = \frac{2x^3 - 5x^2 + 1}{x^2 + x}$. Once obtained:

1. a) Calculate the point where the graph of the function intersects this oblique asymptote.
2. b) Determine, by studying the limit of the difference $[f(x) - (mx+n)]$, whether the function approaches the asymptote from above or from below as $x \to +\infty$.

Given the rational function $f(x) = \frac{2x^3 - 3x^2 + 5}{x^2 - 1}$:

To find the equation of its oblique asymptote $y = mx + n$ and determine the relative position of the curve with respect to it at infinity, two approaches can be followed:

• Method A (Polynomial division): Perform Euclidean division to write $f(x) = (mx + n) + \frac{R(x)}{Q(x)}$.

• Method B (Direct calculation via limits): Find $m = \lim_{x \to \infty} \frac{f(x)}{x}$, then $n = \lim_{x \to \infty} [f(x) - mx]$ and, finally, analyze the sign of the limit of the difference for the position.

1. a) Critically compare both methods: analyze the advantages and disadvantages of each in this case and justify which is more efficient and reliable if the goal is to determine both the equation and the position of the curve relative to the line at both ends ($+\infty$ and $-\infty$).
2. b) Solve the problem applying the method you selected: deduce the equation of the oblique asymptote and determine, with reasoning, whether the graph lies above or below it when $x \to +\infty$ and when $x \to -\infty$.

The real parameters $a$, $b$, $c$, and $k$ are to be determined to define a rational function from the family:

such that it simultaneously satisfies the following three conditions:

1. The line $y = 3$ is a horizontal asymptote of the graph.

2. The graph has exactly one vertical asymptote, located at the line $x = 2$.

3. The function intersects the x-axis at the point $(-1, 0)$.

1. a) Deduce step by step the numerical values of $a$, $b$, $c$, and $k$. Justify with particular rigor why the denominator vanishes at two distinct real values yet the function admits only a single vertical asymptote, explaining what happens geometrically at the other value.
2. b) Based on degree theory and the definition of infinite branches, prove that it is impossible for a function of this type (quotient of two second-degree polynomials) to have an oblique asymptote, regardless of the numerical values its coefficients take.