The Concept of a Limit

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:

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

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) $\lim_{x \to 5} (7)$

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.

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 $x = 3$.

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

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

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 $x = 1$?

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) $f(2)$

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.

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.

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.

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.

The total cost (in pesos) of producing $x$ units of a product is given by the function $C(x) = 300x + 40\,000$. The average cost function is $\bar{C}(x) = \frac{C(x)}{x}$.

1. a) Write 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?

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: $\lim_{x \to +\infty} g(x)$.

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):

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

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?

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.

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 pesos) 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 pesos) follows the function $P(t) = \frac{3\,000t + 4\,000}{t + 1}$, where $t$ is the time in years since its release.

1. a) Analyze 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 $P(t) = \frac{k \times t^2}{2t^2 - 50}$, 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 behavior 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.