Limits at Infinity

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

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

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

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

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

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

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

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

An electronic components recycling company estimates that the total cost (in thousands of dollars) to process $x$ tons of material is given by the function:

$$C(x) = \frac{12x + 45}{3x + 2}$$

What value does the cost per ton tend to when the amount of processed material is extremely large? Interpret the result in terms of "economies of scale".

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 their account. Draw a sketch of the situation and calculate how many followers they will have in the long term (when $t \to +\infty$).

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 the time increases indefinitely?
2. b) Interpret the result in terms of horizontal asymptotes.

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 if the limit when $x \to -\infty$ coincides with the limit when $x \to +\infty$.

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

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

Calculate the limit of the following function when the variable $x$ tends to $+\infty$. To solve it, first identify the dominant term in both the numerator and the denominator:

$$L = \lim_{x \to +\infty} \frac{\sqrt{9x^4 + 2x^2 - 1}}{2x^2 - 5x + 3}$$

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

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

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

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

That is, calculate $\lim_{x \to -\infty} f(x)$. Pay special attention to the sign of the variable $x$ when introducing or extracting factors from the square root and when removing the modulus.

Find the value 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.

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 slope is $m = 3$ and that the limit of the function when $x$ tends to $+\infty$ after subtracting its slope 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. $\lim_{x \to +\infty} [f(x) - 3x] = 4$

Calculate the value of the following limit expression, resolving the indeterminate form $\infty - \infty$ by simplifying to a single fraction and studying the degrees of the polynomials:

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

Once the simple rational form is obtained, indicate whether the function presents a horizontal or oblique asymptote based on the result of the limit.

Given the functions $f(x) = 2^x$ and $g(x) = x^{10} + 500x^2$, reason which of the two functions grows faster when $x \to +\infty$ 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 value according to the order of magnitude of the infinities involved when $x \to +\infty$:

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 fake news on a social network is being analysed. 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 when $x$ 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 the relationship with the number $e$. You must transform the expression so that the form of the definition of this constant appears:

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

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 to correctly apply the formula for resolving the indeterminate form of type number $e$.

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

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

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

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

Calculate the limiting energy consumed by the engine when the speed tends to infinity. Interpret graphically whether the consumption stabilises or continues to grow indefinitely by calculating the horizontal asymptote if it exists.

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 it, consider using the conjugate expression and analyzing 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 precise use of the conjugate technique and a careful analysis of the dominant power within the radicals.