Asymptotes and infinite branches of 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) $\lim_{x \to +\infty} (4 - x^4)$

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

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

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

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.

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 $x = 4$?

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 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).

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 $x \to +\infty$.

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 $y = 5$.

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.