Exercises: Asymptotes of a function

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

A technical inspection drone follows an approach trajectory 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 center. Calculate the limit of the height when the drone is very close to $x = 3$ feet.

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

Calculate the following direct limits by substitution, which represent the velocity of a particle in feet per second at different instants $t$:

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 in relation 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 population of bacteria (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 the asymptotes (vertical and horizontal) of the function $f(x) = \frac{3x - 6}{x + 1}$. To do this:

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

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

Determine the behavior at infinity of the following polynomial functions, indicating 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 received radiation $x$, justify by calculating limits whether the function has any horizontal asymptote and represent it graphically in a schematic way.

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 at the line $y = 3$. Once the value of $a$ is found, calculate the one-sided limits of the function at its vertical asymptotes and explain the behavior of the graph at those points.

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

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

Observe the following graph of a function $f(x)$ and determine, from the visual information, the value of the following limits and the equations of all its asymptotes:

1. a) $\displaystyle \lim_{x \to -\infty} f(x)$
2. b) $\displaystyle \lim_{x \to 1^-} f(x)$ and $\displaystyle \lim_{x \to 1^+} f(x)$
3. 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, indicate 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 results in an indeterminate form of the type $1^\infty$, using the relationship with 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 if there is a horizontal asymptote as $x$ tends to $+\infty$.
3. c) Investigate what happens to the limit of the function as $x \to -\infty$. Is there any asymptote on that branch?

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

1. a) Find the value of $k$ so that the function has a removable discontinuity at $x = 2$.
2. b) If $k = 9$, study the existence of vertical asymptotes and describe the behavior of the function in their vicinity 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) $\displaystyle \lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}$

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

Knowing that, for the system to be stable under extreme loads, the limit of the response time when 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 using modulus functions to model the trajectory of a surveillance drone:

Study the existence of oblique asymptotes for this function. If they exist, calculate their equations and analyse the position of the curve with respect to these asymptotes at both $+\infty$ and $-\infty$.

A communications satellite follows an orbit whose gravitational perturbation is described by the function $g(x)$. It is required that the satellite does not move indefinitely away from a planned linear trajectory. The function is:

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

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 $(0, 2)$ and whose slope is $3$. Once found, check if the function has any vertical asymptote and justify whether it is removable or not.

In the study of the spread of a computer virus on 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 behavior of the infection by calculating $\lim_{t \to +\infty} N(t)$. Does the function present any horizontal asymptote? Justify your answer by applying the corresponding hierarchies of infinities.

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

Determine the one-sided limits at each of the points where the function is not defined and sketch the behavior of the function near its vertical asymptotes.

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

From the result obtained, indicate 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:

1. Find the value of $k$ so that the function has a removable 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 $x \to +\infty$.