Asymptotes of a Function

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

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

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

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 center. Calculate the limit of the height when the drone is very close to $x = 3$ meters.

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

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

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 $f(x) = \frac{sen(x) - x}{x^2 \times sen(x)}$ has a vertical asymptote at the origin.

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.

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 behavior of the function nearby using one-sided limits.

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 behavior of the graph at those points.

Determine the behavior 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 $x \to -\infty$

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.

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?

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 $x \to \infty$.

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.

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

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?

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

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 behavior 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.

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 $a$.

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 $x \to -\infty$.

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 $-\infty$.

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.

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 $x \to +\infty$.

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 behavior of the function near its vertical asymptotes.

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

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 $x \to +\infty$.