Limits at a Point

Determine the limit of the function $g(x) = \frac{x - 5}{x^2 + 1}$ at the following points:

1. a) $x \to 5$

1. b) $x \to -1$

1. c) $x \to 0$

Calculate the following limits by direct substitution into the expression of the function:

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

1. b) $\lim_{x \to 0} \frac{2x + 5}{x - 3}$

1. c) $\lim_{x \to 4} \sqrt{x + 5}$

A market study indicates that the daily profits of a new coffee shop, in pesos, follow the function $B(t) = t^2 + 5t - 10$, where $t$ is the time in months since its opening. Calculate the limit of the profits as time tends to $3$ months ($t \to 3$).

The download speed of a file (in MB/s) depends on the number of users connected to the network, $n$, according to the function $V(n) = \frac{100}{n + 2}$. Determine what value the download speed approaches as the number of users approaches $18$.

Given the function $f(x) = \frac{x^2 - 1}{x - 1}$, complete the following table of values to approximate the limit of the function as $x$ tends to $1$. What value does the function appear to approach?

A content creator has observed that the growth of followers on their channel during the first hour of a livestream follows the function $f(x) = \frac{x^2 - 25}{x - 5}$, where $x$ is the time in minutes. However, right at minute $x = 5$, the platform suffered a small technical error and did not record data.

Estimate the value that the number of followers approaches as time approaches $5$ minutes by completing the following table of values and calculating the limit:

$\lim_{x \to 5} \frac{x^2 - 25}{x - 5}$

Observe the graph of the function $f(x)$ shown below. From it, determine the value of the following limits and the value of the function at the indicated point:

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 piecewise-defined function:

Calculate the one-sided limits at the boundary point:

1. a) $\lim_{x \to 1^-} f(x)$

1. b) $\lim_{x \to 1^+} f(x)$

Does the limit $\lim_{x \to 1} f(x)$ exist? Justify your answer.

Draw a small sketch representing a function $h(x)$ that simultaneously satisfies these three conditions:

1. $\lim_{x \to 3^-} h(x) = 2$

2. $\lim_{x \to 3^+} h(x) = 2$

3. $h(3) = 4$

Is the function continuous at $x = 3$? Explain the difference between the value of the limit and the value of the function at that point.

The temperature of an electronic component in a cooling process is given by the following piecewise-defined function, where $t$ is the time in seconds:

$T(t) = \begin{cases} t^2 - 2t + 5 & \text{if } t < 3 \\ \frac{2t + 18}{t} & \text{if } t \ge 3 \end{cases}$

Calculate the one-sided limits at the boundary point ($t = 3$) and determine whether the overall limit exists at that point. Justify whether the temperature experiences a sudden jump at that moment.

Observe the graph of the function $f(x)$ shown below and analyse its behavior at the indicated points:

1. a) Determine the one-sided limits at $x = -1$ and check whether $\lim_{x \to -1} f(x)$ exists.
2. b) Determine the one-sided limits at $x = 2$. What type of discontinuity does the function have at this point?

Observe the following graph of the function $g(x)$ and analyse its behavior at the indicated critical points:

From the graph, determine:

1. $\lim_{x \to -2^-} g(x)$ and $\lim_{x \to -2^+} g(x)$. Does the limit at $x = -2$ exist?

2. $\lim_{x \to 0^-} g(x)$ and $\lim_{x \to 0^+} g(x)$. What is the value of $g(0)$?

3. Describe the type of discontinuity presented by the function at each of these two points.

A courier company uses a rate that depends on the package weight $x$ in kilograms. The cost function is:

$C(x) = \begin{cases} 5x + 2 & \text{if } 0 < x \le 2 \\ \sqrt{ax + 12} & \text{if } x > 2 \end{cases}$

Find the value of the parameter $a$ so that the cost does not have a discontinuity at the point $x = 2$, that is, so that the limit $\lim_{x \to 2} C(x)$ exists and equals the value of the function.

Calculate the value of $k$ so that the following limit is a real (finite) number and, once $k$ is found, calculate the value of this limit:

Given the piecewise-defined function that models the marginal production cost of a technology company as a function of the units $x$ (in thousands):

Determine the values of the parameters $a$ and $b$ knowing that the limit of the function at the break point $x = 2$ exists and that this limit is equal to $4$. Reasonably justify whether the function is continuous at that point based on the calculated one-sided limits.

Calculate the value of the following limit by resolving the indeterminate form by factorising polynomials:

$\lim_{x \to -2} \frac{x^3 + 3x^2 + 2x}{x^2 - x - 6}$

A sociological study establishes that a user's level of satisfaction with a social network, $S(t)$, measured from 0 to 100, follows the following function in relation to the time $t$ (in months) they have been registered:

Calculate the limit of satisfaction as time approaches $2$ months and $3$ months. Interpret the result obtained: what physically happens to the user's satisfaction at those moments in time?

Solve the following limit using the conjugate product method to eliminate the radical from the numerator:

$\lim_{x \to 4} \frac{\sqrt{x+5} - 3}{x - 4}$

Calculate the value of the following limit by resolving the indeterminate form using the conjugate and polynomial factorisation:

Explain step by step the simplification process and how you manage to eliminate the factor that makes the denominator zero.

Given the rational function $f(x) = \frac{2x^2 - 8}{x^2 - 4x + 4}$, study its behavior as $x$ approaches $2$.

To do so, calculate the one-sided limits:

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

Explain, based on the results, whether the function has a vertical asymptote or a removable discontinuity at that point.

In a biology laboratory, the concentration of a toxin in a bacterial culture is studied. The concentration $C$ in parts per million (ppm) with respect to the temperature $x$ in degrees Celsius follows the function:

$C(x) = \frac{x^2 - 4x + 3}{x^2 - 1}$

Calculate the limiting concentration as the temperature tends to $1^\circ\text{C}$. Is the concentration defined at exactly that temperature? Justify your answer by analysing the domain and the limit at that point.

Find the value of the one-sided limits of the following function at the point $x = 3$:

Based on your results, reason whether the overall limit exists at that point. What graphical difference is there between approaching from the left and from the right?

The spread of a rumour in a school is being studied. The number of students who know the news after $h$ hours is given by the function:

Calculate and interpret the meaning of the following limits in the context of the problem:

1. $\lim_{h \to 0^+} N(h)$

2. $\lim_{h \to 4^-} N(h)$

Is the mathematical model consistent for any value of $h$? Reason your answer based on the existence of vertical asymptotes.

Consider the rational function $R(x) = \frac{x^2 - m}{x^2 - 5x + 6}$.

1. a) Determine the value of $m$ so that the function has a removable discontinuity at $x = 2$.

1. b) For that value of $m$, calculate $\lim_{x \to 2} R(x)$.

1. c) What happens to the limit of the function at $x = 3$ for the value of $m$ found in part a)? Calculate the one-sided limits at $x = 3$ and describe the behavior of the infinite branches of the function near that line.