Exercises: Infinite limits and limits at infinity

Look at the graph of the function $f(x)$ and determine the value of the one-sided limits as the variable $x$ approaches $x = 2$, that is, $\lim_{x \to 2^-} f(x)$ and $\lim_{x \to 2^+} f(x)$. From your observation, explain whether the overall limit $\lim_{x \to 2} f(x)$ exists.

In the graph, the open circle indicates that this point does not belong to the function, and the filled dot does belong to it.

Complete the following table of values for the function $f(x) = \frac{x^2 - 1}{x - 1}$ and intuitively estimate the value of the limit $\lim_{x \to 1} f(x)$. Explain why, even though the function is not defined at $x = 1$, we can still talk about its limit at that point.

Given the function $f(x) = \frac{4}{x - 3}$, study its behaviour as $x$ approaches $3$ from both sides. To do so, calculate:

1. a) $\lim_{x \to 3^-} f(x)$
2. b) $\lim_{x \to 3^+} f(x)$

State whether the function has any vertical asymptote and write its equation.

Imagine that the daily profit of a new specialty café, in euros, follows the function $B(t) = 2t^2 - 10t + 50$, where $t$ is the time in months since its opening. What will happen to the profit if the café keeps operating indefinitely? Calculate $\lim_{t \to +\infty} B(t)$ and interpret the result in the context of the problem.

Calculate the following limits at infinity for the given polynomial functions. Remember that the term of highest degree is the one that determines the end behaviour of the function:

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

The number of users who visit a second-hand clothing website after $x$ days of an advertising campaign is given by the function $N(x) = \frac{3000x + 100}{x + 2}$.

1. a) How many users visited the website on launch day ($x=0$)?
2. b) If the campaign continues for a long time, what constant value will the number of daily visitors tend towards? Calculate $\lim_{x \to +\infty} N(x)$.

Analyse the degree of the numerator and the denominator to determine the value of the following limits at infinity without carrying out complex operations:

1. a) $\lim_{x \to +\infty} \frac{5x^2 - 3x}{2x^2 + 1}$
2. b) $\lim_{x \to +\infty} \frac{x + 7}{x^2 - 4}$
3. c) $\lim_{x \to +\infty} \frac{x^3 - 1}{100x^2 + 5}$

Match each of the following functions with its behavior at infinity (horizontal asymptote). Justify your answer by calculating the corresponding limit as $x \to +\infty$:

1. $f(x) = \dfrac{2x - 1}{x + 3}$

2. $g(x) = \dfrac{10}{x + 1}$

3. $h(x) = \dfrac{6x^2}{3x^2 - 5}$

Possible behaviors (the same option may correspond to more than one function):

A) Horizontal asymptote $y = 0$

B) Horizontal asymptote $y = 2$

C) Horizontal asymptote $y = 3$

D) No horizontal asymptote

In an Applied Mathematics for Social Sciences II exam, Sergio encounters the calculation of the limit:

Sergio writes in his solution:

"Upon substituting, we obtain $\frac{3 \cdot 2 - 1}{2 - 2} = \frac{5}{0} = +\infty$. Therefore, the limit is $+\infty$ and the line $x = 2$ is a vertical asymptote."

1. a) Explain precisely the conceptual error Sergio makes by directly stating that the two-sided limit is $+\infty$.
2. b) Carry out the correct analysis by separately calculating the one-sided limits as $x \to 2^-$ and as $x \to 2^+$.
3. c) Based on the results of the one-sided limits, conclude whether the overall limit $\lim_{x \to 2} \frac{3x - 1}{x - 2}$ exists or not, and justify whether the final statement about the existence of the vertical asymptote was correct or incorrect.

A Valencian agricultural cooperative estimates that the average cost per kilogram of citrus fruit produced (in euros), when harvesting $x$ tonnes of fruit, is given by the function:

1. a) Estimate: before performing any algebraic manipulation, look at the highest-degree terms in the numerator and denominator. What value do you think the average cost will approach if the harvest grows indefinitely ($x \to +\infty$)? Justify your intuition.
2. b) Numerically evaluate the function for $x = 100$ and $x = 10\,000$ (you may leave the operations set up or round the result to two decimal places). Do these values support the estimate you made?
3. c) Check: formally calculate $\lim_{x \to +\infty} C(x)$ by dividing both numerator and denominator by the highest power of $x$. Based on this procedure, justify why the lower-degree terms ($-12x$, $+9$, $+3x$, and $+5$) turn out to be mathematically irrelevant in determining the limit at infinity.

A market study analyzes the evolution of the number of users of a new messaging app, measured in thousands, by means of the function $U(t) = \dfrac{40t + 10}{2t + 5}$, where $t$ represents the time in months since its launch. The figure shows the graph of this function.

1. a) Analyze what happens to the number of users around the third month. To do so, calculate the following limit, showing your reasoning:

1. b) Determine the long-term behavior of the app. Is there a maximum value that the number of users approaches? Interpret the result in terms of horizontal asymptotes.

Given the rational function $f(x) = \dfrac{x^2 - kx}{x^2 - 4}$:

1. a) Find the value of the parameter $k \in \mathbb{R}$ so that the function has a removable discontinuity at $x = 2$.

1. b) For the value of $k$ found in the previous part, study the behaviour of the function as $x$ tends to $-2$. What type of limit do you obtain? Justify whether a vertical asymptote exists by calculating the one-sided limits:

Calculate the following limits at infinity using the comparison of dominant terms and explain their graphical meaning (whether they tend to a specific value or display parabolic branches):

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

1. b) $\lim_{x \to -\infty} \dfrac{x^2 + 10}{x - 5}$

1. c) $\lim_{x \to +\infty} \dfrac{\sqrt{x^4 + 3x}}{5x^2 - 1}$

A forest surveillance drone follows a path whose height in meters above sea level is given by the expression $H(t) = \sqrt{t^2 + 8t} - t$, where $t$ is the time in minutes since it starts its patrol. The figure shows its path during the first 20 minutes.

Calculate the limit of the height as time tends to infinity:

At what height does the drone level off its flight in the long run? Resolve the indeterminate form by multiplying by the conjugate.

Consider the function $g(x) = \dfrac{2x + 1}{x - 3}$.

1. a) Complete the following table of values to analyse the trend of the function as $x$ approaches $3$ from the right:

1. b) From the results, deduce the value of the one-sided limit $\lim_{x \to 3^+} g(x)$.

1. c) Calculate the limit $\lim_{x \to +\infty} g(x)$ and explain the conceptual difference between the "limit at a point" and the "limit at infinity" for this function.

Given the graph of the function $h(x)$ shown below, identify with reasons the value of the following limits:

1. a) $\lim_{x \to -1^-} h(x)$ and $\lim_{x \to -1^+} h(x)$
2. b) $\lim_{x \to 2} h(x)$
3. c) $\lim_{x \to +\infty} h(x)$
4. d) $\lim_{x \to -\infty} h(x)$

Calculate the values of $a$ and $b$ so that the following limit equalities hold:

1. a) $\lim_{x \to +\infty} \dfrac{ax^2 + 5x}{3x^2 - 2} = 4$

1. b) $\lim_{x \to 1} \dfrac{x^2 + bx - 3}{x - 1} = 4$ (Note: For the limit to be finite at a point that makes the denominator zero, the numerator must also be zero).

In a biotechnology laboratory, the population of a strain of bacteria (in millions) grows according to the model:

where $t \ge 0$ is the time in hours.

1. a) What is the initial population of bacteria ($t=0$)?

1. b) Calculate $\lim_{t \to +\infty} P(t)$. What does this value represent for the colony of bacteria?

1. c) Compare the order of infinity of the numerator and the denominator in the fraction. If the growth were $100t^3$ instead of $100t^2$, would the long-term behaviour change? Justify your answer using limits.

Based on statistical series from Red Eléctrica de España (REE), a team of analysts models the percentage of peninsular electricity generation from renewable sources, $P(t)$ (in percent), as a function of the years elapsed since 2010 ($t = 0$ represents the year 2010):

1. a) Find the percentage of renewables predicted by the model for the year 2010 ($t = 0$) and for the year 2020 ($t = 10$).
2. b) Calculate $\lim_{t \to +\infty} P(t)$ and interpret the result: does the function have any horizontal asymptote? From the perspective of the energy sector, does this model foresee that Spain could be $100\,\%$ supplied solely by renewable energy in the long term? Justify your answer.
3. c) A more ambitious European directive proposes an alternative model: $Q(t) = \frac{96t^2 + 500}{t^2 + 2t + 10}$. Calculate its limit as $t \to +\infty$ and compare the long-term theoretical bound of both energy transition projections.

Design a rational function of the form $f(x) = \frac{a x^2 + b x}{x^2 - c}$ knowing that it must simultaneously satisfy three conditions imposed by an economic model:

1. It has a horizontal asymptote at the line $y = 2$ as $x \to +\infty$.

2. It has a vertical asymptote at the line $x = 3$.

3. At $x = -3$ it does not have a vertical asymptote, but a removable discontinuity; that is, the limit $\lim_{x \to -3} f(x) = L$ exists and is finite.

1. a) Determine, with reasoned justification, the values of the real parameters $a$, $b$, and $c$.
2. b) With the values obtained, calculate the value of the finite limit $L = \lim_{x \to -3} f(x)$.
3. c) Conceptually explain why a vertical asymptote is generated at $x = 3$ while a removable discontinuity occurs at $x = -3$, relating it to whether or not the numerator and denominator vanish simultaneously.

Use the formal definition of a limit based on neighbourhoods to prove that the limit of the function $f(x) = \frac{3x - 1}{2}$ as $x$ tends to $3$ is equal to $4$. To do so, given an $\varepsilon > 0$, find an expression for $\delta$ in terms of $\varepsilon$ that satisfies the condition:

$0 < |x - 3| < \delta \implies |f(x) - 4| < \varepsilon$

Consider the rational function $f(x) = \frac{ax^2 + bx + 2}{x^2 - 4}$. Determine the values of the real parameters $a$ and $b$ knowing that the function has a horizontal asymptote at $y = 2$ and that a finite limit exists at the point $x = 2$. Once the parameters are found, calculate the value of this limit at $x = 2$ and discuss the behaviour of the function as $x$ tends to $-2$.

Calculate the value of the following limit involving a difference of irrational functions at infinity:

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

Explain with reasons why we cannot simply say that the result is $0$ when subtracting two infinities of the same order.

In a biotechnology laboratory, the growth of a bacterial strain in a closed environment is modelled by the function $P(t) = \frac{100 \cdot e^{0,5t}}{k + e^{0,5t}}$, where $P(t)$ is the population in millions and $t$ is the time in hours.

1. a) Knowing that the initial population ($t=0$) is 20 million, calculate the value of the constant $k$.
2. b) Determine the limit of the population as time tends to infinity and interpret the biological meaning of the horizontal asymptote obtained.

Find the values of $k$ so that the following limit based on the structure of the number $e$ is equal to $e^6$:

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

Show all intermediate steps, including resolving the indeterminate form $1^\infty$.

Given the piecewise function:

$f(x) = \begin{cases} \frac{\ln(1+x^2)}{x} & \text{if } x < 0 \\ \frac{a \cdot \sin(x)}{x} + b & \text{if } x > 0 \end{cases}$

Calculate the values of $a$ and $b$ so that the one-sided limits exist at $x=0$, the limit from the left is equal to $0$ and the limit from the right is equal to $5$. Is it possible to define $f(0)$ so that the function is continuous at that point?

Analyze the asymptotic behavior of the function $g(x) = \frac{2^{x+1} + 5}{2^x - 8}$. To do so:

1. a) Calculate the limits as $x \to +\infty$ and as $x \to -\infty$. What horizontal asymptotes does it have?

1. b) Identify the point where the denominator vanishes and study the one-sided limits at that point to determine whether there is a vertical asymptote.

Calculate the value of the following limit, comparing orders of infinity and resolving the resulting indeterminate form:

$\lim_{x \to +\infty} \frac{x^{100} + 3^x}{\log_2(x^5) + e^x}$

Justify your answer based on the hierarchy of growth of exponential, polynomial (power) and logarithmic functions. Would the result change if the limit were towards $-\infty$? Reason your answer.

To calculate the limit $\lim_{x \to +\infty} \left(\sqrt{x^2 + 6x} - x\right)$, three second-year Bachillerato students propose different approaches:

• Álvaro's proposal: "Since for very large values of $x$ the term $6x$ is negligible compared to $x^2$, I approximate $\sqrt{x^2 + 6x} \approx \sqrt{x^2} = x$. Therefore, the limit is $\lim_{x \to +\infty} (x - x) = 0$."

• Beatriz's proposal: "I factor out $x$ from the square root: $x \cdot \left(\sqrt{1 + \frac{6}{x}} - 1\right)$. Upon substituting, I get $\infty \cdot (\sqrt{1} - 1) = \infty \cdot 0$, and since any number multiplied by zero is zero, the limit is $0$."

• Carlos's proposal: "The expression presents an indeterminate form of type $\infty - \infty$ between terms of the same order. We must multiply and divide by the conjugate expression $\sqrt{x^2 + 6x} + x$ and then compare the resulting degrees."

1. a) Analyze Álvaro's and Beatriz's proposals. Explain with mathematical rigor what mistake each one makes and why their justifications are flawed.
2. b) Evaluate the limit by applying the method proposed by Carlos, detailing each step of the calculation.
3. c) If the limit to be evaluated were $\lim_{x \to +\infty} \left(\sqrt{9x^4 + 2x} - 2x\right)$, would it be necessary to use the conjugate method or would it suffice to compare the orders of growth of both terms? Choose the most efficient method and justify your choice without needing to carry out the entire conjugate calculation.

The technology department of a Spanish banking institution models the average cost per operation (in euro cents) when processing a volume of $x$ million transactions using the function:

In parallel, the average server response time for complex queries (in milliseconds) is estimated using the function:

1. a) Apply the hierarchy of growth at infinity ($\log_a x \ll x^n \ll a^x$) to calculate $\lim_{x \to +\infty} f(x)$ in terms of the parameters $a$ and $b$. If the bank requires the long-term average cost to stabilize at exactly $4$ cents per operation, what must the value of $a$ be? Does the value of parameter $b$ influence this long-term behavior? Justify your answer.
2. b) Suppose that a code optimization removes the exponential component from the numerator ($a = 0$, keeping $b > 0$). In this case, calculate $\lim_{x \to +\infty} f(x)$ using the hierarchy of infinites and interpret what would happen to the unit cost as the transaction volume grows indefinitely.
3. c) Determine the value of the real parameter $k$ so that the response time function $g(x)$ has the line $y = 0,6$ as a horizontal asymptote. Justify how the term $\ln(x)$ is handled compared to the polynomial term as $x \to +\infty$.