Exercises: Concept

A marketing analyst is studying the growth in followers of an Instagram account. The function $f(x)=3x$ represents the number of followers (in thousands) $x$ hours after a campaign is launched. To understand what happens right at the fourth hour, complete the following table of values by approaching $x = 4$ and predict the value of the limit $\lim_{x \to 4} f(x)$.

1. a) Calculate the missing values of $f(x)$ in the table (you can give up to three decimal places).

1. b) Observe which value $f(x)$ gets closer to as $x$ approaches $4$ from the left and from the right, and predict the value of $\lim_{x \to 4} f(x)$.

Look at the following graph, which represents the charge level of a smart battery $C(t)$ (in %) as a function of time $t$ (in minutes). Due to a software error, the sensor does not record the exact value at minute $t = 3$ (hollow point), although the charge seems to follow a clear trend. Use the graph to determine the following values:

1. a) $\lim_{t \to 3^-} C(t)$ (limit as we approach 3 from the left).
2. b) $\lim_{t \to 3^+} C(t)$ (limit as we approach 3 from the right).
3. c) Does the two-sided limit $\lim_{t \to 3} C(t)$ exist? Justify your answer.

For a function $f(x)$ to be continuous at a point $x = a$, three fundamental mathematical conditions must be met. Write down what these three conditions are and check whether the function $f(x) = x^2 - 1$ is continuous at the point $x = 2$ by verifying each of them step by step.

A delivery drone follows a trajectory whose height in meters is given by the function $h(t)$. At a given moment in the flight, the drone makes a sharp altitude adjustment. The function is defined piecewise as follows:

$h(t) = \begin{cases} 2t + 5 & \text{if } 0 \leq t < 4 \\ 15 - t & \text{if } t \geq 4 \end{cases}$

Calculate the one-sided limits at the instant $t = 4$:

1. a) $\lim_{t \to 4^-} h(t)$
2. b) $\lim_{t \to 4^+} h(t)$
3. c) In light of the results, state whether the drone changes height continuously or if it makes a "jump" at that instant.

Explain in your own words the difference between the value of a function at a point, $f(a)$, and the limit of the function as $x$ approaches that point, $\lim_{x \to a} f(x)$. Use the concept of "closeness" to help write your answer.

Given the rational function $f(x) = \frac{x + 5}{x - 3}$, identify the point where the function is not defined. Calculate the limit as $x$ approaches that value from the right ($\lim_{x \to 3^+} f(x)$) by giving $x$ values very close to it, such as $3.1$, $3.01$ and $3.001$. Towards what value (finite or infinite) does the function seem to tend?

In real life, many situations are continuous and others have jumps. Classify the following situations as Continuous or Discontinuous, briefly justifying your choice from the perspective of the concept of limit:

1. a) The growth in a teenager's height over a year.
2. b) The cost of sending a package by courier according to its weight (if the price changes abruptly for each full kilogram).
3. c) The temperature of a room from the moment the air conditioning is switched on.
4. d) The balance of a bank account as card payments are made.

Given the function shown graphically below, state with reasons whether it is continuous at the points $x = -2$, $x = 0$ and $x = 2$. If it is not continuous at any point, explain which of the three conditions of continuity fails.

Given the function $f(x)$ shown in the following graph, analyse the existence of the limit at the critical points. To do this:

1. State the value of the one-sided limits when $x \to -1$ and when $x \to 2$.

2. Determine whether the overall limit exists at those points.

3. Explain the difference between the value of the function $f(2)$ and the value of $\lim_{x \to 2} f(x)$.

A market study indicates that the profit of a new mobile app (in thousands of pesos) follows the function $B(t) = \frac{40t^2 - 160}{t - 2}$, where $t$ is the time in months since its launch.

1. What happens to the profit as the time approaches 2 months? Make a numerical approximation by calculating $B(1,9)$, $B(1,99)$, $B(2,01)$, and $B(2,1)$.

2. Although the function is not defined at $t = 2$, could you assign a logical value to the profit at that instant based on the trend? Justify your answer using the concept of limit.

Determine the value of parameter $k \in \mathbb{R}$ so that the function $f(x)$ is continuous at the point $x = 3$. Once found, explain whether the function has any type of discontinuity if $k = 5$.

$f(x) = \begin{cases} \frac{x^2 - k}{x + 1} & \text{if } x < 3 \\ \sqrt{x + 1} & \text{if } x \geq 3 \end{cases}$

Analyse the behavior of the function $g(x) = \frac{x + 3}{x^2 - 9}$ near its vertical asymptotes.

1. Identify the points where the function is not continuous.

2. Calculate the one-sided limits at $x = 3$ and $x = -3$.

3. Classify the discontinuities found (removable, infinite jump or finite jump).

The growth of a user community on a social network has been modeled by the function $P(t) = \frac{50t + 100}{t + 4}$, where $P$ is the number of users (in thousands) and $t$ is time in years.

1. How many users does the social network have at the initial moment ($t = 0$)?

2. Calculate the limit of $P(t)$ as $t \to +\infty$. What is the meaning of this value in the context of the problem?

3. Illustrate the situation using a diagram that shows the upper bound of the growth.

Consider the piecewise function:

$h(x) = \begin{cases} e^x & \text{if } x \leq 0 \\ \frac{1}{x+k} & \text{if } x > 0 \end{cases}$

1. Find the value of $k$ so that the function is continuous at $x = 0$.

2. If $k = 1$, calculate $\lim_{x \to -\infty} h(x)$ and $\lim_{x \to +\infty} h(x)$. Does the function have any horizontal asymptote?

Given the function $f(x) = \frac{|x-1|}{x-1}$:

1. Calculate the one-sided limits at the point $x = 1$.

2. Based on the previous results, reason whether the overall limit exists at $x = 1$.

3. Is it possible to redefine the function at $x = 1$ so that it is continuous? Justify your answer based on the type of discontinuity.

At a courier company, the shipping cost $C(x)$ (in pesos) of a parcel of $x$ kg is given by the following function:

$C(x) = \begin{cases} 5 + 2x & \text{if } 0 < x \leq 5 \\ 12 + 0.5x & \text{if } 5 < x \leq 10 \\ 20 & \text{if } x > 10 \end{cases}$

1. a) Study the continuity of the cost function at the points $x = 5$ and $x = 10$.

1. b) A customer claims that "for a one-gram difference in the weight of the parcel, the price sometimes makes an unjustified jump". Is he right? Identify at which weights this happens and calculate the size of each "jump".

A mobile app development company has modeled the number of daily active users, $U(t)$, in thousands, of a new social network using the following function, where $t$ represents the time in months since its launch:

$U(t) = \begin{cases} t^2 + at + 10 & \text{if } 0 \le t \le 4 \\ \ln(t - 3) + b & \text{if } t > 4 \end{cases}$

It is known that in month $t = 2$ the app reached a local minimum of $6000$ users ($U(2) = 6$). Determine the values of parameters $a$ and $b$ so that the function is consistent with the concept of continuity at the moment of the change in marketing strategy ($t = 4$). Once found, state the real domain of the model if the app cannot have fewer than $1000$ users to remain operational.

Analyse the domain of definition of the following complex function, which combines logarithmic and rational constraints under a square root:

$f(x) = \sqrt{\frac{\ln(x^2 - 8)}{x - 5}}$

Express the domain using interval notation. Reason how each component (the logarithm, the denominator and the root) affects the exclusion of values on the real line.

A smart air-conditioning system regulates the cooling power $P$ (in kW) according to the outside temperature $T$ (in $^\circ C$) following the function:

$P(T) = | |T - 25| - 5 |$

1. a) Graph the function on the interval $T \in [15, 35]$ (it will help you to first make a table of values and pay attention to the points where the trend changes).

1. b) From the graph and the concept of function, determine for which temperatures the power supplied is exactly $2$ kW.

1. c) Explain the physical meaning, in this context of energy saving, of the points where the trend changes (the "peaks" and "valleys" of the graph, where the power goes from increasing to decreasing or vice versa).

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

Given the functions $f(x) = \frac{x + 1}{x - 2}$ and $g(x) = \sqrt{x^2 - 9}$, find:

1. a) Determine the domain of the composite function $h(x) = (g \circ f)(x)$.
2. b) Check whether there is any value of $x$ for which the image under $h(x)$ is equal to $0$.
3. c) Justify, based on the definition of image, whether the value $y = 1$ belongs to the range of the function $f(x)$.

The profit function of an agricultural cooperative depends on the investment in technology $x$ (in thousands of pesos) and is given by (with $B(x)$ also in thousands of pesos):

$B(x) = \frac{20x^2 - 80}{x^2 + 4}$

Carry out a complete study of its characteristics as a function:

1. a) Find the axis intercepts and explain their meaning in the context of the problem (what happens if there is no investment? How much investment is needed to start making a profit?).
2. b) Determine the theoretical maximum value that the profit could reach if the investment grew indefinitely (horizontal asymptote).
3. c) State whether the function is symmetric and what implication this would have if the domain allowed negative investments (loans).

Consider the function $f(x) = \frac{e^x}{e^x - 2}$.

1. a) Calculate its domain of definition.
2. b) Study its behavior when $x$ takes very large positive values ($x \to +\infty$) and very large negative values ($x \to -\infty$).
3. c) Find the inverse function $f^{-1}(x)$ and specify the domain of this new function. How is the domain of $f^{-1}(x)$ related to the range of $f(x)$?

The growth of a lynx population in a nature park is modeled by the function:

$N(t) = \frac{500}{1 + 9e^{-0.2t}}$

Where $N$ is the number of lynxes and $t$ is the time in years. The figure shows the graph of this function.

1. a) Calculate the initial population ($t = 0$).

1. b) Prove that the function is always increasing by analyzing the structure of the expression.

1. c) Is there a physical limit to the park's population? Justify your answer using the concept of long-term behavior.

Let $f(x)$ be a piecewise function representing the bank interest applied to a deposit according to the capital $x$ invested:

$f(x) = \begin{cases} \frac{kx}{x+2\,000} & \text{if } 0 \le x \le 10\,000 \\ \frac{5\,000}{x-9\,000} & \text{if } x > 10\,000 \end{cases}$

1. a) Determine the value of $k$ so that there is no sudden jump in the interest rate when going from investing $\$10\,000$ to investing a slightly higher amount.
2. b) Once $k$ is found, analyse whether there is any capital value $x > 10\,000$ where the function ceases to be defined. Would that point make economic sense in this model?
3. c) What is the image of $x=0$ and what interpretation does it have in the banking context?