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.

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 $x = 1$?

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

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) $f(2)$

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.

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?