Exercises: Concept

Imagine you are analysing the growth of a new social media account. The number of followers in thousands, after $x$ weeks, is given by the function $f(x)$. To understand what happens near the third week, we have obtained the following values:

$x$	2.9	2.99	2.999	...	3.001	3.01	3.1
$f(x)$	5.81	5.98	5.998	...	6.002	6.02	6.19

By observing the table, intuitively estimate the value of the following limit:

$$\lim_{x \to 3} f(x)$$

Given the following graph of a function $f(x)$, determine the value of the one-sided limits at the point of discontinuity $x = 2$:

1. a) $\lim_{x \to 2^-} f(x)$
2. b) $\lim_{x \to 2^+} f(x)$
3. c) Does the overall limit $\lim_{x \to 2} f(x)$ exist? Justify your answer.

Observe the graph of the function $g(x)$ and analyse what happens as the independent variable $x$ takes larger and larger values. What value does the function seem to approach in the "long run"?

Express your conclusion using limit notation: $\lim_{x \to +\infty} g(x)$.

Consider the piecewise-defined function:

$$f(x) = \begin{cases} x + 3 & \text{if } x < 1 \\ 5 - x & \text{if } x \ge 1 \end{cases}$$

Calculate the one-sided limits at $x = 1$ by substituting directly into the corresponding expression:

1. a) $\lim_{x \to 1^-} f(x)$
2. b) $\lim_{x \to 1^+} f(x)$
3. c) In light of the results, is the function continuous at $x = 1$?

Sometimes, the value of a function at a point does not coincide with the value that the function approaches near that point. Observe the following graph and determine:

1. a) The value of the function at the point: $f(3)$
2. b) The value of the limit: $\lim_{x \to 3} f(x)$

Calculate the following limits of polynomial and constant functions using basic properties (direct substitution):

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

The temperature of an electronic component that heats up when turned on is being studied. The temperature $T$ (in $^\circ C$) with respect to time $t$ (in seconds) follows the function $T(t) = \frac{80t}{t+2}$.

Calculate the value of the temperature as time becomes "infinite" (that is, the stabilisation temperature):

$$\lim_{t \to +\infty} \frac{80t}{t+2}$$

Look at the graph of the function $h(x)$. The marked points correspond to values of $x$ that get closer and closer to $0$ from the right.

1. a) Describe in your own words what happens to the values of $h(x)$ when $x$ gets close to $0$ from the right.

1. b) Express that result using limit notation.

Carla is studying the behaviour of the piecewise function:

$$f(x) = \begin{cases} 2x + 1 & \text{if } x < 2 \\ 8 & \text{if } x = 2 \\ x^2 + 1 & \text{if } x > 2 \end{cases}$$

In an exercise asking to calculate $\lim_{x \to 2} f(x)$, Carla writes the following reasoning in her notebook:

"I calculate the one-sided limits: $\lim_{x \to 2^-} f(x) = 2 \cdot 2 + 1 = 5$ and $\lim_{x \to 2^+} f(x) = 2^2 + 1 = 5$. Since both one-sided limits are equal to $5$, but the value of the function at that point is $f(2) = 8$, I conclude that the limit $\lim_{x \to 2} f(x)$ does not exist, because for it to exist it must necessarily coincide with the value of the function at that point."

1. a) Precisely identify the conceptual error made by Carla and explain the difference between the existence of the limit of a function at a point and the value that said function takes at that same point.
2. b) Correct Carla's conclusion: determine with reasoning whether the limit $\lim_{x \to 2} f(x)$ exists and, if so, what its value is according to the definition of a limit.

Consider the rational function $g(x) = \frac{3x - 6}{x^2 - 4}$, whose domain does not include the value $x = 2$.

1. a) Before operating algebraically, estimate intuitively what value $g(x)$ approaches as $x$ approaches $2$. To do this, calculate using a calculator the numerical values of $g(1,99)$ and $g(2,01)$, rounding to four decimal places, and formulate a conjecture about the value of the limit $\lim_{x \to 2} g(x)$.
2. b) Factor the numerator and the denominator to simplify the expression of $g(x)$ when $x \neq 2$, and calculate the limit $\lim_{x \to 2} g(x)$ rigorously.
3. c) Compare your estimation from part a with the result obtained in part b. Reason why it is possible for the limit to be a finite real number if attempting to substitute $x = 2$ directly into the original function results in division by zero.

A new strategy video game estimates that the number of active users, measured in thousands, follows the function $U(t) = \frac{20t + 50}{t + 1}$, where $t$ is the time in months since its launch.

1. a) Complete the following table of values to observe the trend in the number of users as the months pass:

$t$	10	100	1 000	10 000
$U(t)$

1. b) Use the results in the table to estimate the value of $\lim_{t \to +\infty} U(t)$.

1. c) What does the result obtained mean in the context of the problem? Will the game eventually run out of users?

Observe the graph of the function $f(x)$ shown below and determine, if they exist, the values of the indicated limits. If any limit does not exist, briefly justify why based on the one-sided limits.

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

Given the following piecewise-defined function, determine the value of the parameter $k$ so that the limit of the function exists at the split point $x = 3$.

$f(x) = \begin{cases} x^2 - kx + 2 & \text{if } x < 3 \\ \frac{x+9}{x-1} & \text{if } x \ge 3 \end{cases}$

A popular funfair attraction opens its doors to the public. The number of people per minute queuing at the entrance during the first few moments is modelled by the function $P(t) = \frac{100}{t^2 - 8t + 16}$, where $t$ is the time in minutes.

1. a) Analyse what happens to the influx of visitors as time approaches $t = 4$ minutes by calculating the one-sided limits $\lim_{t \to 4^-} P(t)$ and $\lim_{t \to 4^+} P(t)$.

1. b) Is it possible for the attraction to continue admitting visitors at that rate after 4 minutes? Justify your answer based on the concept of an infinite limit.

Consider the function $g(x) = \frac{x^2 - 9}{x - 3}$.

1. a) Try to calculate $g(3)$. What happens?

1. b) Create a table of values for $x$ very close to 3 from the left ($2,9; 2,99; 2,999$) and from the right ($3,1; 3,01; 3,001$).

1. c) In light of the results, what is the value of $\lim_{x \to 3} g(x)$?

1. d) Graphically represent the situation, highlighting what exactly happens at the point $x = 3$.

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

In a biology laboratory, the population of a bacteria colony is studied after applying an antibiotic. The population (in millions) is given by $P(t) = 5 + e^{-0,2t}$, where $t$ is time.

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

1. b) Calculate the limit of the population as time tends to infinity: $\lim_{t \to +\infty} P(t)$.

1. c) Draw a sketch of the graph of the function based on the points you have calculated and its asymptotes.

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

It is said that the limit of a function at a point is the value that the outputs approach, regardless of whether the function is defined at that point or not. Give reasons why the following statements are true or false:

1. a) If $f(a)$ exists, then $\lim_{x \to a} f(x)$ always exists.
2. b) If $\lim_{x \to a} f(x) = L$ exists and $f(a)$ exists, then necessarily $f(a) = L$.
3. c) For the limit at a point to exist, the one-sided limits must be equal and finite.

The total cost (in euros) of producing $x$ units of a product is given by the function $C(x) = 15x + 2000$. The average cost function is $\bar{C}(x) = \frac{C(x)}{x}$.

1. a) Write down the expression for the function $\bar{C}(x)$.

1. b) Calculate $\lim_{x \to 0^+} \bar{C}(x)$ and interpret the result. What happens if we manufacture very few units?

1. c) Calculate $\lim_{x \to +\infty} \bar{C}(x)$. What value does the manufacturing cost per unit tend to when production is on a massive scale?

An eco-friendly urban delivery courier company using electric vans in Zaragoza offers a service for SMEs with a monthly fee. The total monthly cost in euros is given by a fixed fee of $150$ € for insurance and maintenance, plus $0,20$ € per kilometre travelled. Therefore, the average cost per kilometre (in euros per kilometre) when $x$ kilometres are travelled in a month is modelled by the function:

$$C(x) = \frac{150 + 0,20x}{x}, \quad x > 0$$

1. a) Calculate $\lim_{x \to 0^+} C(x)$ and interpret its practical meaning for a business owner who signs up for the service but barely travels any kilometres in a month. What geometric element does this result represent on the graph of $C(x)$?
2. b) Calculate $\lim_{x \to +\infty} C(x)$ and interpret the economic outcome: what happens to the impact of the fixed fee as the monthly mileage becomes very large?
3. c) Determine the equation of the horizontal asymptote of the function $C(x)$ and reason whether the average cost per kilometre could ever reach $0,18$ € in any month if a huge number of kilometres were travelled.

You must construct the analytical expression of a piecewise function, $f(x)$, that simultaneously satisfies the following three conditions:

1. The limit from the left as $x$ approaches $1$ must exist and be finite: $\lim_{x \to 1^-} f(x) = 4$.

2. The vertical line $x = 1$ must be a vertical asymptote from the right tending to positive infinity: $\lim_{x \to 1^+} f(x) = +\infty$.

3. The horizontal line $y = -2$ must be a horizontal asymptote towards negative infinity: $\lim_{x \to -\infty} f(x) = -2$.

To achieve this:

1. a) Propose a specific formula for each branch of the function $f(x)$, clearly defining the intervals of its domain.
2. b) Demonstrate analytically, by calculating the corresponding limits, that the function you designed satisfies each of the three required conditions exactly.

Use the formal definition of the limit of a function at a point to prove rigorously that:

$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$$

To do this, given a value of $\epsilon > 0$, find an expression for $\delta$ in terms of $\epsilon$ such that the implication of the definition holds. Explain the algebraic reasoning step by step.

Observe the following graph representing the evolution of the profits of a tech startup (in thousands of euros) as a function of time $t$ (in months). Determine with reasons, from observing the trends, the values of the following limits and explain their meaning in the context of the problem:

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

Calculate the values of the real parameters $a$ and $b$ so that the function $f(x)$ has a limit at the split points $x = -1$ and $x = 3$. Once found, state whether the function is continuous at those points.

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

In a market study, it has been determined that the price of a collector's item (in euros) follows the function $P(t) = \frac{150t + 200}{t + 1}$, where $t$ is the time in years since its release.

1. a) Analyse the long-term price trend by calculating the limit as $t \to +\infty$.
2. b) If the model were to change after a crisis in year $t=10$ to $P(t) = \frac{k \cdot t^2}{2t^2 - 50}$, what value should $k$ have so that the price does not experience a sudden jump (that is, so that the limit exists at $t=10$)?

Consider the functions $f(x) = \frac{1}{x-3}$ and $g(x) = x^2 - 9$. Study the behaviour of the limit of the composite function $(f \circ g)(x)$ as $x$ tends to $2$ and as $x$ tends to $\sqrt{12}$.

State whether the limits are finite or infinite by analysing the one-sided limits in each case.

Given the function $f(x) = \frac{x^2 - kx + 6}{x^2 - 9}$:

1. a) Find the value of $k$ such that the function has a removable discontinuity at $x = 3$.
2. b) For that value of $k$, calculate the value of the limit $\lim_{x \to 3} f(x)$.
3. c) What happens to the limit of the function at $x = -3$ for that same value of $k$? Justify the existence of a vertical asymptote by calculating the one-sided limits.

A biologist observes that the population of an insect colony in a laboratory (in hundreds of individuals) behaves according to the function $N(t)$, where $t$ is the time in weeks:

$$N(t) = \frac{2t^2 - 8}{t^2 + t - 6}$$

1. a) The biologist notices that something strange happens near the second week. Calculate $\lim_{t \to 2} N(t)$ and interpret whether the population disappears, stabilises, or grows indefinitely at that moment.
2. b) Determine the initial population of the colony ($t=0$) and the population to which the colony will tend if conditions remain constant indefinitely.

Draw an approximate graphical sketch of a function $f(x)$ that simultaneously satisfies all of the following limit conditions. Then, propose a possible algebraic expression (piecewise or rational) for that function.

1. $\lim_{x \to -\infty} f(x) = 2$

2. $\lim_{x \to -2^{-}} f(x) = +\infty$ and $\lim_{x \to -2^{+}} f(x) = -\infty$

3. $\lim_{x \to 0} f(x) = 0$, but $f(0) = 1$

4. $\lim_{x \to +\infty} f(x) = +\infty$ (parabolic branch)

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

We want to study the existence and, if applicable, the value of the limit:

$$\lim_{x \to 3} \frac{|x - 3|}{x^2 - 9}$$

Three 11th-grade students (1.º de Bachillerato) propose different strategies to approach it:

• Marcos's method (numerical): He proposes using a calculator to substitute a value very close to $3$, for example $x = 2,999$, calculating the obtained result and stating that this result matches the sought limit.

• Laura's method (direct substitution): She proposes directly substituting $x = 3$, obtaining $\frac{|0|}{0} = \frac{0}{0}$, and immediately concludes that the limit does not exist because it is an indeterminate form lacking a numerical value.

• Sara's method (one-sided limits): She proposes splitting the function by removing the absolute value according to the definition of $|x - 3|$ (for $x < 3$ and for $x > 3$), simplifying the resulting algebraic fraction on each branch, and calculating the one-sided limits $\lim_{x \to 3^-}$ and $\lim_{x \to 3^+}$ separately.

1. a) Critically evaluate Marcos's method: why can approaching $3$ using only values from the left lead to a serious error in this particular case?
2. b) Critically evaluate Laura's method: why does the appearance of the expression $\frac{0}{0}$ not allow ruling out the existence of the limit?
3. c) Apply Sara's method: calculate both one-sided limits step by step and deduce, justifying it with the formal definition of a limit at a point, whether $\lim_{x \to 3} \frac{|x - 3|}{x^2 - 9}$ exists or not.

The National Statistics Institute (INE) is monitoring a pilot digitalisation programme in a rural region of Spain. The estimated proportion of elderly people connected to the internet (as a percentage of the total elderly population of said region), after $t$ months from the start of the plan, is described by the function:

$$P(t) = \begin{cases} \frac{k \cdot t^2 + 10t}{t + 2} & \text{if } 0 \le t < 4 \\[2mm] \frac{60t - 40}{t + 1} & \text{if } t > 4 \end{cases}$$

where $k$ is a real parameter that depends on the initial allocation of instructors and technological resources.

1. a) To prevent the model from showing an abrupt and incoherent jump at the end of the fourth month, the limit $\lim_{t \to 4} P(t)$ is required to exist. Calculate the value that the parameter $k$ must take for this to happen, justifying your procedure in detail using one-sided limits.
2. b) Imagine that the regulatory body determines that the specific census data collected exactly at the fourth month is $P(4) = 50\,\%$. With the value of $k$ obtained in part a, does the limit $\lim_{t \to 4} P(t)$ still exist? Reason your answer, explaining why the existence of the limit at $t = a$ is conceptually independent of the point value $P(a)$ or of whether it is even defined.
3. c) Calculate $\lim_{t \to +\infty} P(t)$, determine the horizontal asymptote line, and interpret the result in sociological terms: will the entirety of the elderly population of the region eventually be connected according to the long-term forecasts of this model?