Exercises: Limits at a point

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 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 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 ($t \to 3$).

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

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 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.

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) $x \to 0$

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.

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:

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

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.

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 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.

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

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

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?

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.

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 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.

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 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?

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?

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:

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.