Asymptotes of a function

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).

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?

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?

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 $-\infty$.

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 behaviour of the function near its vertical asymptotes.

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

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 $a$.

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.