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 euros, 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.

Marcos has to calculate the following limit for a homework assignment: $\displaystyle\lim_{x \to 2} \frac{x - 2}{x^2 - 4}$. In his notebook, he writes the following work:

1. a) Precisely identify the algebraic error Marcos makes in the second step. What expression actually remains in the numerator when dividing both numerator and denominator by $(x - 2)$?
2. b) Explain why the result $4$ makes no sense by observing that for values very close to $2$ (for example, $x = 2,1$), the numerator is much smaller than the denominator.
3. c) Correct the procedure and calculate the correct value of the limit.

Given the function $f(x) = \frac{x^2 + 2x - 3}{x - 1}$, we want to find $\displaystyle\lim_{x \to 1} f(x)$.

1. a) Before applying any algebraic technique, estimate the value of the limit by calculating the numerical values of the function for $x = 0,99$ and for $x = 1,01$. Toward which integer does $f(x)$ seem to approach?
2. b) Justify why direct substitution of $x = 1$ does not allow you to immediately answer what the limit is.
3. c) Evaluate the limit algebraically by factoring the numerator and simplifying. Check whether the obtained value matches your initial estimate and explain why canceling the factor $(x - 1)$ is valid when calculating a limit even though it is not valid for $x = 1$.

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 behaviour 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 behaviour 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.

Two 1st-year Bachillerato students are debating how to solve $\displaystyle\lim_{x \to 3} \frac{x^2 - 2x - 3}{x^2 - 6x + 9}$. Upon substituting $x = 3$, both observe that the form $\frac{0}{0}$ is obtained.

• Lucía claims: "Since it is an indeterminate form $\frac{0}{0}$, the remainder theorem guarantees that we can simplify the factor $(x - 3)$ and we will obtain a finite real number, that is, a removable discontinuity."

• Sergio argues: "Getting $\frac{0}{0}$ does not ensure that the limit is finite. We have to factor and check the multiplicity of the root $x = 3$ in the denominator; if after simplifying the denominator still vanishes, there will be a vertical asymptote and the limit will not exist as a finite number."

1. a) Determine which of the two arguments is correct and explain the conceptual error in the other person's statement.
2. b) Evaluate the limit by factoring the numerator and denominator, simplify the expression, and calculate the one-sided limits $\displaystyle\lim_{x \to 3^-} f(x)$ and $\displaystyle\lim_{x \to 3^+} f(x)$ to conclude whether the requested limit exists or not.

Design the algebraic formula for a rational function $f(x) = \frac{P(x)}{Q(x)}$, where both $P(x)$ and $Q(x)$ are second-degree polynomials with integer coefficients, that simultaneously satisfies these three conditions:

1. It yields an indeterminate form $\frac{0}{0}$ when evaluated directly at $x = 2$, but its limit at that point exists and is finite: $\displaystyle\lim_{x \to 2} f(x) = 5$.

2. It has a vertical asymptote at the line $x = -1$.

3. It passes through the origin, that is, $f(0) = 0$.

Explain the reasoning that allows you to deduce the factors of $P(x)$ and $Q(x)$ from each condition, find the value of the necessary constants, and write the final expression of $f(x)$ with its polynomials expanded.

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 behaviour 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.

An olive oil cooperative in Jaén estimates that the total daily cost of bottling $x$ hundreds of liters of extra virgin olive oil is given by $C(x) = x^2 + 6x + 40$ (in euros, for $x \geq 0$). The finance department needs to assess the marginal cost upon reaching a bottling rate of $400$ liters ($x = 4$). To do so, they study the average rate of change of the cost between a generic production level $x$ and the base level $x = 4$:

1. a) Calculate $C(4)$ and write the expression for $T(x)$. Justify which indeterminate form appears when directly evaluating $T(4)$ and explain the economic meaning of both numerator and denominator vanishing simultaneously when there is no change in production.
2. b) Resolve the indeterminate form by factoring to find the exact marginal cost by calculating $\displaystyle\lim_{x \to 4} T(x)$ (in euros per each additional hundred liters).
3. c) A competing oil mill uses a different technology whose incremental rate follows the function $G(x) = \frac{\sqrt{x} - 2}{x - 4} + 14$. Calculate $\displaystyle\lim_{x \to 4} G(x)$ by multiplying by the conjugate and determine, with reasoning, which of the two cooperatives incurs a lower marginal cost upon reaching $400$ liters.

Consider the function $f(x) = \frac{\sqrt{x + 7} - k}{x - 2}$, where $k$ is a real constant.

1. a) Reason why the limit $\displaystyle\lim_{x \to 2} f(x)$ can only be a finite real number if substituting $x = 2$ produces the indeterminate form $\frac{0}{0}$. Use this reasoning to deduce the only permissible value of $k$.
2. b) For the value of $k$ obtained, calculate $\displaystyle\lim_{x \to 2} f(x)$ analytically by multiplying by the conjugate expression and simplifying the common factor.
3. c) If, instead of the value found, we set $k = 5$, justify theoretically (without needing to calculate both full one-sided limits) whether the line $x = 2$ will represent a removable discontinuity or a vertical asymptote, and state with reasoning the sign toward which the function will tend as $x \to 2^+$.