Exercises: Limits at infinity x → ±∞

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.

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

An electronics recycling company estimates that the total cost (in thousands of euros) to process $x$ tonnes of material is given by the function:

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

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

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

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.

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

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

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:

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

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:

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

Marcos is solving the following limit at infinity:

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

On his exam, he writes the following justification: "Since the highest polynomial degree that appears in both the numerator and the denominator is 3, we look at the leading coefficients of $x^3$; therefore, the limit is $\frac{5}{4}$".

1. a) Explain in detail why Marcos's reasoning is conceptually flawed, indicating which families of functions are involved in the expression and what the hierarchy of infinities is among them as $x \to +\infty$.
2. b) Solve the limit correctly, justifying which term dictates the growth rate in the numerator and which one in the denominator.

Consider the function $f(x) = \left(\frac{x+3}{x+1}\right)^{2x}$. We want to study its asymptotic behavior as $x \to +\infty$.

1. a) Make a reasoned qualitative estimation before applying formulas: observe that for large values of $x$, the base $\frac{x+3}{x+1}$ is strictly greater than 1 but very close to 1 (for instance, for $x = 100$, it is $\frac{103}{101} \approx 1{,}0198$), while the exponent $2x$ grows without bound. Do you intuitively expect the limit to be 1, less than 1, or greater than 1? Justify your answer.
2. b) Formally solve the limit $\lim_{x \to +\infty} f(x)$ by identifying the type of indeterminate form and applying the formula $\lim g(x)^{h(x)} = e^{\lim h(x)\cdot(g(x)-1)}$.
3. c) Express the exact value obtained in terms of the number $e$ and check whether your initial estimation was correct, taking into account that $e \approx 2{,}718$.

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 $x \to +\infty$.

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

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

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.

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

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

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:

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

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

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:

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

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.

The National Statistics Institute (INE), in its reports on territorial digitization, analyzes ultra-fast broadband coverage in rural municipalities with fewer than 5.000 inhabitants in Spain. The estimated percentage of rural households with access to this network as a function of time is modeled by the function:

$$P(t) = \frac{88t^2 + 120t}{t^2 + 2t + 5}$$

where $t$ is the elapsed time in years since the implementation of the national strategy ($t = 0$, corresponding to the year 2018).

1. a) Calculate the percentage of rural households that had a connection at the start of the plan ($t = 0$) and interpret the result.
2. b) Determine and mathematically justify the value of $\lim_{t \to +\infty} P(t)$ by comparing the degrees of the numerator and the denominator, and explain what practical meaning this limit has for territorial cohesion.
3. c) An operator claimed in the press that this model guarantees that, over the years, universal coverage ($100\,\%$) will be achieved in these rural areas. Based on the analytical results obtained, refute or corroborate this claim with mathematical arguments.

To calculate the limit at infinity:

$$\lim_{x \to +\infty} \left( \sqrt{x^4 + 6x^2} - x^2 \right)$$

two students propose different strategies:

• Method A (Marta): Factor out the highest power from inside the root: $\sqrt{x^4\left(1 + \frac{6}{x^2}\right)} - x^2 = x^2\sqrt{1 + \frac{6}{x^2}} - x^2 = x^2\left(\sqrt{1 + \frac{6}{x^2}} - 1\right)$. Since $\frac{6}{x^2} \to 0$, she substitutes $\sqrt{1+0}-1 = 0$ and concludes that the result is $\infty \cdot 0 = 0$.

• Method B (Sergio): Identify an indeterminate form of the type $\infty - \infty$ between terms of the same order and multiply and divide by the conjugate expression $\sqrt{x^4 + 6x^2} + x^2$.

1. a) Critically analyze Method A: explain why it is not mathematically valid to substitute only part of the expression by its partial limit while keeping another part variable, and recall the nature of the expression $\infty \cdot 0$.
2. b) Carry out Method B in detail and obtain the correct value of the limit.
3. c) Based on the comparison of both methods, establish a general criterion: under what conditions is it essential to use the conjugate instead of merely comparing the dominant terms of each term?

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.

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).

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 $e$.

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.

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.

Calculate the value of the following limit expression, resolving the indeterminacy $\infty - \infty$ 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, state whether the function has a horizontal or oblique asymptote based on the result of the limit.

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.

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

Find the values of the real parameters $a$, $b$, and $c$ such that the function:

$$f(x) = \sqrt{4x^2 + ax + 1} - (bx + c)$$

simultaneously satisfies the following two asymptotic conditions:

1) $\lim_{x \to +\infty} f(x) = 0$

2) $\lim_{x \to -\infty} \left( f(x) + 4x \right) = 3$

To solve the problem in an orderly manner:

1. a) Analyze condition (1) as $x \to +\infty$: deduce with reasoning what constant value $b$ must take so that the limit is not infinite and, applying the conjugate technique, obtain a linear relationship between $a$ and $c$.
2. b) Analyze condition (2) as $x \to -\infty$: perform the formal change of variable $x = -t$ (with $t \to +\infty$), taking into account that $\sqrt{x^2} = |x| = -x = t$, apply the corresponding technique, and obtain a second equation with $a$ and $c$.
3. c) Solve the resulting system, write the final analytic expression of the function $f(x)$, and check the consistency of the results.

Carlos needs to calculate the following limit at negative infinity:

$$\lim_{x \to -\infty} \left( \sqrt{x^2 - 4x} + x \right)$$

In his notebook, he develops the following steps:

1. He applies the change of variable $x = -t$ with $t \to +\infty$: $\lim_{t \to +\infty} \left( \sqrt{(-t)^2 - 4(-t)} + (-t) \right) = \lim_{t \to +\infty} \left( \sqrt{t^2 + 4t} - t \right)$.

2. He reasons: "As $t \to +\infty$, the term $4t$ is negligible compared to $t^2$, so $\sqrt{t^2 + 4t} \approx \sqrt{t^2} = t$".

3. He concludes: "Therefore, the limit is $\lim_{t \to +\infty} (t - t) = 0$".

His teacher tells him that step 1 is correct, but that the final conclusion is false and that the actual limit is 2.

1. a) Point out the conceptual error made in step 2: explain why approximating the square root solely by its highest-degree term leads to a critical loss of information when subtracting infinities of the same order.
2. b) Correct the calculation by solving the limit resulting from step 1 using the rigorous conjugate procedure until justifying obtaining the value 2.
3. c) Carlos asks: "Why in a quotient like $\lim_{t \to +\infty} \frac{\sqrt{t^2 + 4t}}{3t}$ is it perfectly valid to neglect the $4t$ and approximate the root by $t$ to give $\frac{1}{3}$, whereas in the subtraction it is not allowed?". Answer Carlos by distinguishing with mathematical rigor the behavior of asymptotic approximations in quotients versus additions or subtractions with cancellation.