Limits at Infinity

Calculate the limit of the following polynomial function as the independent variable increases indefinitely:

$$\lim_{x \to +\infty} (-3x^4 + 5x^2 - 10x + 1)$$

Explain which term you based your result on and why the other terms do not affect the value of the limit.

Determine the value of the following limits based on comparing the degrees of the polynomials:

1. a) $\lim_{x \to +\infty} \frac{5x^2 - 3x + 8}{2x^2 + 1}$

1. b) $\lim_{x \to -\infty} \frac{x^3 + 2x}{x^4 - 5}$

Given the rational function $f(x) = \dfrac{ax^2 + 3x - 5}{2x^2 - k}$, calculate the values of the real parameters $a$ and $k$ knowing that the function has a horizontal asymptote at $y = 3$ and that its graph passes through the point $(1, -2)$. Once found, check whether the limit as $x \to -\infty$ coincides with the limit as $x \to +\infty$.

Look at the graph of the function $f(x)$ and determine visually the value of the limits at infinity. Also state the equation of the horizontal asymptote, if it exists.

1. a) $\lim_{x \to +\infty} f(x)$
2. b) $\lim_{x \to -\infty} f(x)$

The number of followers of a new influencer on a social network, measured in thousands, evolves according to the function:

$$S(t) = \frac{100t^2}{2t^2 + 50}$$

where $t$ is the time elapsed in months since opening the account. Draw a sketch of the situation and calculate how many followers they will have in the long term (when $t \to +\infty$).

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

A video game server has estimated that the number of simultaneously connected users during the first year follows the function $U(t) = \dfrac{500t^2 + 1000}{0.2t^2 + 5}$, where $t$ is the time in months.

1. a) What happens to the number of users if time increases indefinitely?
2. b) Interpret the result in terms of horizontal asymptotes.

Calculate the limit of the following rational function containing radicals, simplifying the expression using effective powers:

$$\lim_{x \to +\infty} \frac{\sqrt{9x^2 + 5}}{2x - 1}$$

Calculate the limit as $x \to -\infty$ of the following function. Pay special attention to the signs when the negative variable leaves or enters an even-degree root:

$$\lim_{x \to -\infty} \frac{3x - 2}{\sqrt{4x^2 + 1}}$$

Study the limit at negative infinity of the following rational function containing an absolute value and a square root:

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

That is, calculate $\lim_{x \to -\infty} f(x)$. Take special care with the sign of the variable $x$ when inserting or extracting factors from the square root and when removing the absolute value.

Given the functions $f(x) = 2^x$ and $g(x) = x^{10} + 500x^2$, reason which of the two functions grows faster as $x \to +\infty$ using the scale of infinities. From your answer, deduce the value of the following limit:

$$\lim_{x \to +\infty} \frac{x^{10} + 500x^2}{2^x}$$

Given the functions $f(x) = e^x$, $g(x) = x^{10}$, and $h(x) = \ln(x)$, order the following limits from smallest to largest according to the order of magnitude of the infinities involved as $x \to +\infty$:

1. $L_1 = \lim_{x \to +\infty} \dfrac{g(x)}{f(x)}$

2. $L_2 = \lim_{x \to +\infty} \dfrac{g(x)}{h(x)}$

3. $L_3 = \lim_{x \to +\infty} \dfrac{h(x)}{f(x)}$

The spread of a piece of fake news on a social network is being analyzed. The number of affected users $N(t)$ after $t$ hours is given by the function:

$$N(t) = \frac{e^{0.5t} + 10t^4}{2e^{0.5t} + \ln(t^2 + 1)}$$

Determine the long-term behavior of the function, that is, calculate $\lim_{t \to +\infty} N(t)$. Justify your answer based on the comparison of orders of infinity (exponential versus polynomial and logarithmic).

Complete the following table of values for the function $f(x) = \left( \frac{x+1}{x} \right)^x$ and use the results obtained to conjecture which famous number the function tends to as $x$ becomes very large.

$x$	$10$	$100$	$1000$	$10000$
$f(x)$	$2.5937$	$2.7048$	$2.7169$	$2.7181$

Write the result using limit notation:

$$\lim_{x \to +\infty} \left( 1 + \frac{1}{x} \right)^x = \dots$$

Determine the value of the following limit using its relation to the number $e$. You must transform the expression so that it takes the form of the definition of this constant:

$$\lim_{x \to +\infty} \left( \frac{2x + 5}{2x - 1} \right)^{x+3}$$

Find the value of the parameter $k \in \mathbb{R}$ so that the following limit of type $1^\infty$ is equal to $\sqrt[3]{e^2}$:

$$\lim_{x \to +\infty} \left( \frac{3x^2 + kx - 1}{3x^2 + 2} \right)^{\frac{x^2 + 1}{x}}$$

Make sure you correctly apply the formula for resolving indeterminate forms of the type involving the number $e$.

Solve the following limit involving a difference of irrational functions. Remember to use the conjugate method to resolve the indeterminate form $[\infty - \infty]$:

$$\lim_{x \to +\infty} \left( \sqrt{x^2 + 5x} - \sqrt{x^2 - 3x} \right)$$

Calculate the values of the real parameters $a$ and $b$ so that the following equality holds at infinity:

$$\lim_{x \to +\infty} \left( \sqrt{4x^2 + ax + 1} - \sqrt{4x^2 - 3x + b} \right) = \frac{5}{4}$$

To solve this, consider using the conjugate expression and analysing the dominant terms in the resulting numerator and denominator.

Calculate the limit of the sequence of functions given by the following nested root:

$$\lim_{x \to +\infty} \left( \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x} \right)$$

This exercise requires a precise use of the conjugate technique and a careful analysis of the dominant power within the radicals.

Find the values of the parameters $a$ and $b$ so that the following equality holds:

$$\lim_{x \to +\infty} \left( \frac{x^2 + 1}{x - 1} - ax - b \right) = 0$$

This type of limit is fundamental for calculating oblique asymptotes, which you will study in later topics.

A prototype ion propulsion engine for satellites has an energy consumption $E(x)$ in joules that depends on the speed $x$ (in km/s) according to the expression:

$$E(x) = \sqrt{x^2 + 4x + 2} - x$$

The figure shows the graph of $E(x)$ for $0 \le x \le 20$.

Calculate the limiting energy that the engine consumes as the speed tends to infinity. Interpret graphically whether the consumption stabilizes or keeps growing indefinitely by calculating the horizontal asymptote, if it exists.

Given the function $f(x) = \frac{ax^3 + bx^2 + 1}{2x^2 - 4}$, determine the values of the real coefficients $a$ and $b$ knowing that the function has an oblique asymptote whose gradient is $m = 3$ and that the limit of the function as $x$ tends to $+\infty$ after subtracting its gradient component (that is, the term $n$ of the asymptote $y=mx+n$) is $4$.

In other words, solve for $a$ and $b$ knowing that:

1. $\lim_{x \to +\infty} \frac{f(x)}{x} = 3$

2. $\lim_{x \to +\infty} [f(x) - 3x] = 4$