Asymptotes and infinite branches

Given the rational function $f(x) = \frac{3x - 6}{x + 2}$, carry out the following study:

1. a) Identify its vertical asymptote.

1. b) Calculate the limit of the function as $x$ tends to $+\infty$ to find its horizontal asymptote.

Calculate the vertical and horizontal asymptotes of the function $f(x) = \frac{2x + 1}{x - 4}$, writing down the limits that justify them.

True or false? Reason your answer with an example: "Every rational function has at least one vertical asymptote."

The download speed of a file is being monitored in Megabytes per second (MB/s). The function describing the speed after $t$ seconds is:

$v(t) = \frac{10t}{t + 5}$

Find the infinite branches by calculating the limit as $t \to +\infty$. What value does the download speed stabilise at in the long term?

Consider the function $f(x) = e^x + 2$.

• a) Does it have vertical asymptotes? Why or why not?

• b) Calculate $\lim_{x \to -\infty} f(x)$ and deduce its horizontal asymptote.

• c) Is the graph above or below this asymptote? And what happens when $x \to +\infty$?

Given the rational function $f(x) = \frac{ax+b}{x-3}$, determine the values of the real parameters $a$ and $b$ given that:

• The function has a horizontal asymptote at $y = 2$.

• The graph of the function passes through the point $P(1, -1)$.

Once $a$ and $b$ are found, study the monotonicity of the function and check whether it has any stationary points.

Study the infinite branches and the existence of asymptotes of the function $f(x) = \frac{x^2 - 1}{x+2}$.

1. Find the equation of the oblique asymptote.

2. Determine the position of the curve relative to this asymptote as $x \to +\infty$ and as $x \to -\infty$.

3. Find the points on the graph where the tangent line is parallel to the oblique asymptote.

4. Draw an approximate sketch of the asymptotes and the function.

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

Study the infinite branches of the function $f(x) = \frac{3x^2 - 2}{x + 1}$: vertical asymptote, oblique asymptote, and position of the curve relative to the latter as $+\infty$ and as $-\infty$.

Consider the function $f(x) = \frac{x^2 - 4}{x - 1}$.

• a) Determine its domain and its vertical asymptote, studying the one-sided limits.

• b) Find the oblique asymptote and study the position of the curve relative to it as $+\infty$ and as $-\infty$.

• c) Calculate the axes intercepts and sketch the complete graph with its asymptotes.

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

A sociological study on the spread of a rumour in a student community follows the function $P(t) = 100(1 - e^{-0,2t})$, where $P$ is the percentage of people who know the rumour and $t$ is the time in hours ($t \ge 0$).

1. Calculate the initial percentage of people who know the rumour.

2. What value does the percentage of people approach as time passes? Interpret the result in terms of asymptotes.

3. Justify, using the first derivative, that the rumour always spreads (the function is increasing).

4. Does the curvature of the function change at any point? Reason your answer.

The average production cost of an independent magazine is $C(x) = \frac{500 + 2x}{x}$ euros per copy, where $x$ is the number of copies printed.

• a) Find the horizontal asymptote of $C(x)$ and interpret it: why can the average cost not fall below a certain value?

• b) What happens to the average cost when very few copies are printed? Relate this to the vertical asymptote.

• c) Does it make sense to speak of an oblique asymptote for this function? Justify your answer.

Analyze and graph the function $f(x) = \dfrac{e^x}{x^2 - 3x}$. Your study must include:

1. Domain and points of discontinuity.

2. Infinite limits and vertical asymptotes.

3. Horizontal asymptotes and parabolic branches (studying the behavior both at $+\infty$ and at $-\infty$).

4. Calculation of the first derivative to find intervals of monotonicity and relative extrema.

5. Sketch of the graph indicating all the elements you found (do it by hand in your notebook or with the tool you prefer).