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 behavior 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 pesos, 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 behavior 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

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 behavior 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 flight 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 began its patrol.

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

At what height does the drone stabilize its flight in the long term? 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 behavior change? Justify your answer using limits.

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 behavior 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 $P(t) = \frac{100 \times e^{0.5t}}{k + e^{0.5t}}$, 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:

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

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.