Exercises: One-sided limits

Given the function $f(x) = x^2 - 3$, we want to investigate what happens when the values of $x$ get very close to $2$. Complete the following table of values and, in light of the results, state what you think the value of $\lim_{x \to 2} f(x)$ is.

Observe the following graph of the function $g(x)$ and determine the value of the one-sided limits at the boundary point $x = 1$. Also state whether the overall limit $\lim_{x \to 1} g(x)$ exists.

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

Explain in your own words what the expression $\lim_{x \to a^-} f(x) = L$ means. Give an everyday example where an approximation to a value "from the left" occurs (for example, the time remaining before an exam starts or the money left on a travelcard).

Calculate the following simple limits using direct substitution:

1. a) $\lim_{x \to 3} (2x^2 - 5x + 1)$
2. b) $\lim_{x \to -1} \frac{x + 5}{x + 2}$
3. c) $\lim_{x \to 0} \sqrt{x + 4}$

Given the piecewise function:

$f(x) = \begin{cases} x + 2 & \text{if } x < 3 \\ 7 - x & \text{if } x \ge 3 \end{cases}$

Calculate the one-sided limits at the point $x = 3$:

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

What can you say about the existence of the limit $\lim_{x \to 3} f(x)$?

The charge level of an electric scooter (as a percentage) after $t$ hours of use behaves according to the function $C(t) = 100 - 15t$. If a user wants to know how much battery will be left when the fourth hour of the journey is about to be reached, calculate the limit:

$\lim_{t \to 4^-} (100 - 15t)$

Look at the graph of this function $h(x)$, which has a vertical asymptote at $x = 2$ (the dashed line), and determine the value of the indicated one-sided limits.

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

Consider the function $f(x)$ shown in the following graph. The empty circle indicates a point that does not belong to the graph and the filled point indicates the value the function takes at $x = 2$.

From the graph:

1. a) Identify the value of $f(2)$.

1. b) Identify $\lim_{x \to 2^-} f(x)$.

1. c) Identify $\lim_{x \to 2^+} f(x)$.

1. d) Does the limit of $f(x)$ as $x \to 2$ exist? Justify your answer.

To determine whether $\lim_{x \to 3} f(x)$ exists for the piecewise function

Marcos states in class: "Since for $x = 3$ the function equals $8$, then the overall limit $\lim_{x \to 3} f(x)$ is directly equal to $8$, without any need to calculate one-sided limits". Answer the following questions:

1. a) Identify and explain, with reasoning, the conceptual error that Marcos makes regarding the concept of a limit and its relationship with the pointwise value $f(3)$.
2. b) Rigorously calculate the one-sided limits $\lim_{x \to 3^-} f(x)$ and $\lim_{x \to 3^+} f(x)$.
3. c) From your results, determine whether $\lim_{x \to 3} f(x)$ exists, state its value if it does, and compare it with the value of the function at that point.

Consider the function $f(x) = \frac{2x - 4}{|x - 2|}$, defined for all $x \neq 2$.

1. a) Before operating analytically, estimate the behavior of $f(x)$ near $2$: mentally calculate the sign and value you expect to obtain if you substitute values very close to $2$ from the left (such as $x = 1,99$) and from the right (such as $x = 2,01$). Write down your prediction for $\lim_{x \to 2^-} f(x)$ and $\lim_{x \to 2^+} f(x)$.
2. b) Rewrite the function as a piecewise function by eliminating the absolute value according to the sign of $x - 2$, and formally calculate both one-sided limits.
3. c) Check whether your initial estimate matched the calculations and conclude with reasoning whether the overall limit $\lim_{x \to 2} f(x)$ exists.

Given the graph of the function $f(x)$ shown below, analyse the behaviour of the function near the critical points. Determine, with justification, the one-sided limits and the overall limit (if it exists) for:

1. a) $x \to -2$
2. b) $x \to 1$
3. c) $x \to 3$

An investment manager observes that the yield of a high-risk fund, $R(x)$, expressed in thousands of euros based on the invested capital $x$ (in tens of thousands), follows the model:

$R(x) = \frac{x^2 - 9}{x - 3}$

Investigate, by completing a table of values using a calculator, what happens to the yield as the investment approaches $30.000$ € ($x \to 3$). Use the values $2,9; 2,99; 2,999$ from the left and $3,1; 3,01; 3,001$ from the right. Could the manager claim that the limit exists even though the function is not defined at $x = 3$? Justify your answer.

Determine the value of the parameter $k$ so that the limit of the following function exists at its boundary point $x = 2$:

$f(x) = \begin{cases} kx - 3 & \text{if } x < 2 \\ x^2 + k & \text{if } x \geq 2 \end{cases}$

For the limit to exist, remember that the one-sided limits must be equal. Once $k$ is found, state the value of the limit.

A logistics company charges for express deliveries using a tiered rate based on weight $p$ in kilograms. The cost $C(p)$ in euros is given by:

$C(p) = \begin{cases} 5 + 2p & \text{if } 0 < p \leq 5 \\ 10 + 1,5p & \text{if } p > 5 \end{cases}$

Calculate and interpret in the context of the problem:

1. a) $\lim_{p \to 5^-} C(p)$
2. b) $\lim_{p \to 5^+} C(p)$
3. c) Is there a continuously defined cost for a parcel weighing exactly $5$ kg?

Consider the function $g(x) = \frac{2x + 1}{x - 4}$.

1. a) Study the one-sided limits of $g(x)$ as $x \to 4$.

1. b) State whether the function has a vertical asymptote at that point.

1. c) Draw a small sketch (by hand, with the line $x = 4$ marked) showing where the branches of the function tend (towards $+\infty$ or towards $-\infty$) as you approach $x = 4$ from the left or from the right.

Given the modulus function $h(x) = \frac{|x - 1|}{x - 1}$:

1. a) Express the function as a piecewise function by removing the modulus sign.
2. b) Calculate $\lim_{x \to 1^-} h(x)$ and $\lim_{x \to 1^+} h(x)$.
3. c) Does the overall limit $\lim_{x \to 1} h(x)$ exist? Justify your answer based on the theory of one-sided limits.

In a sociological study, the level of alienation $A(t)$ of a community after $t$ months of a regulatory change is defined by:

$A(t) = \begin{cases} \frac{t^2}{4} & \text{if } 0 \leq t < 6 \\ 12 - \frac{t}{2} & \text{if } t \geq 6 \end{cases}$

Investigate whether the level of alienation changes abruptly or gradually in the sixth month. To do so, calculate the one-sided limits at $t = 6$ and check whether the limit exists.

Given the function $f(x) = \frac{x^2 - 1}{x^2 - 2x + 1}$, calculate the one-sided limits at the point where the denominator vanishes.

1. a) What happens to the values of the function as we approach $x = 1$ from the left?

1. b) And from the right?

1. c) Based on your results, can we say that the overall limit at $x = 1$ exists?

Hint. Factor the numerator and the denominator and simplify the fraction before studying the sign on each side.

An express parcel delivery company in Spain applies a fee that depends on the weight $p$ of the package (in kilograms, with $0 < p \le 5$). The cost in euros, $C(p)$, is given by:

where $k$ is a parameter to be set by commercial management.

1. a) Explain the economic meaning of the one-sided limits $\lim_{p \to 2^-} C(p)$ and $\lim_{p \to 2^+} C(p)$ for a customer preparing a shipment of around $2\text{ kg}$.
2. b) Calculate the value that $k$ must take so that there is no abrupt change (jump) in the fee around $2\text{ kg}$; that is, so that $\lim_{p \to 2} C(p)$ exists. What is the limiting cost of the package in that case?
3. c) If the company mistakenly set $k = 0,2$, calculate the new one-sided limits at $p = 2$ and interpret what commercial problem or paradox would arise between sending a $1,99\text{ kg}$ package and a $2,01\text{ kg}$ package.

Design the analytic expression of a piecewise function $g(x)$, with domain all of $\mathbb{R}$, that simultaneously satisfies these four conditions:

1. $\lim_{x \to -1^-} g(x) = 4$ and $\lim_{x \to -1^+} g(x) = 4$.

2. $g(-1) = 1$.

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

4. On each open interval $(-\infty, -1)$, $(-1, 3)$, and $(3, +\infty)$, the function must be a straight line (a polynomial of degree less than or equal to 1).

Once you have proposed the formula for your function:

1. a) Prove, by calculating the one-sided limits, that your function satisfies all the required conditions exactly.
2. b) Explain, with reasoning, at which of the two points ($x = -1$ or $x = 3$) the overall limit exists and why its existence is not possible at the other.

The evolution of the profits of a tech startup (in thousands of euros) is being studied as a function of time $t$ in months. The function describing the profit is given by:

Determine the values of the parameters $a$ and $b$ knowing that the overall limit of the function as $t \to 2$ exists. Once found, calculate and interpret the one-sided limits at $t = 2$.

Calculate the one-sided limits at the break points of the following function, which combines a modulus with a rational function:

From the results obtained, reason whether $\lim_{x \to 3} g(x)$ exists and describe the behaviour of the function near $x=3$ and $x=1$.

A courier company establishes the shipping cost $C(x)$ (in euros) for packages of weight $x$ (in kg) using the following function, which uses the "floor" function $E[x]$ (the greatest integer less than or equal to $x$):

Calculate the one-sided limits of the cost function as the weight approaches $2$ kg. Is the cost a continuous function at that point? Justify your answer using the definition of limit.

Consider the piecewise function $f(x)$ defined as:

Given that $\lim_{x \to 0^-} f(x) = 4$, determine the value of the constant $k$, showing your reasoning. Then, find the value of $a$ so that the limit of the function at the origin exists.

(Hint: You can use that $\lim_{h \to 0} \dfrac{e^h - 1}{h} = 1$).

Let the function be $h(x) = \frac{\sqrt{x^2 + k} - 3}{x-1}$.

1. a) Determine the value of $k \in \mathbb{R}$ such that the function gives an indeterminate form of the type $0/0$ when $x \to 1$.
2. b) For this value of $k$, calculate the one-sided limits at $x=1$ and investigate whether the overall limit exists.

In a stock market, the price of a stock $P$ as a function of the risk $x$ is modelled as:

$P(x) = \frac{ax^2 + bx}{x^2 - 1}$

It is known that when the risk approaches $1$ from the right, the price tends to $+\infty$, but when it approaches $1$ from the left, the price tends to $-\infty$. Furthermore, when the risk tends to $+\infty$, the price tends to $1$, and for a risk $x = 2$, the price is $2$. Calculate the values of $a$ and $b$ and check the consistency of the one-sided limits at $x = 1$ and $x = -1$.

Given the function $f(x) = \frac{x^2 - |x|}{x}$, investigate whether the limit at $x=0$ exists by calculating its one-sided limits. Graph the situation and explain what happens to the image of the function exactly at the origin.

You can do it on paper and send a photo with your answer.

Study the one-sided limits of the following function at the points $x = -2$ and $x = 2$. At which of them does a finite limit exist?

Subsequently, propose a modification to the numerator so that at the point where the limit was infinite, it becomes a specific real value.

We want to study the existence of $\lim_{x \to 2} \frac{|x^2 - 4|}{2x - 4}$. Two 2nd-year Bachillerato students propose different strategies to solve it:

• Alba's method: She first studies the sign of the quadratic polynomial $x^2 - 4$ across the entire real line, writes the complete function split into three pieces according to the resulting intervals, and, from that general expression, calculates the one-sided limits at $x = 2$.

• Bruno's method: He reasons locally, solely in a neighborhood of $x = 2$. He argues that if $x \to 2^+$, then $x > 2$ and therefore $x^2 - 4 > 0$, which allows replacing $|x^2 - 4|$ with $(x^2 - 4)$; similarly, if $x \to 2^-$, then $x < 2$ (close to $2$) and therefore $x^2 - 4 < 0$, replacing it with $-(x^2 - 4)$. Then he evaluates each one-sided limit independently by factoring the $\frac{0}{0}$ indeterminate form.

1. a) Apply both methods in detail to calculate $\lim_{x \to 2^-} f(x)$ and $\lim_{x \to 2^+} f(x)$, deducing whether the overall limit exists or not.
2. b) Critically evaluate both approaches: which one is more efficient if we are only interested in the behavior at $x = 2$? In what mathematical context (for example, if graphing or an overall continuity study on $\mathbb{R}$ were required) would Alba's method become essential?

Consider the real function $f(x)$ defined by:

1. a) Analytically study the behavior of the function to the right of $x = 5$. Justify how the absolute value is eliminated for $x > 5$ and calculate $\lim_{x \to 5^+} f(x)$. What value would have been obtained if the expression $\frac{|x - 5|}{x^2 - 25}$ had been used to calculate an approximation from the left ($x \to 5^-$)?
2. b) Calculate the one-sided limit $\lim_{x \to 2^-} f(x)$ by resolving the corresponding indeterminate form.
3. c) Set up and solve the system of equations that determines the values of the parameters $m$ and $n$ so that the overall limits $\lim_{x \to 2} f(x)$ and $\lim_{x \to 5} f(x)$ exist simultaneously.
4. d) Suppose the additional condition is imposed that the middle piece is horizontal ($m = 0$). Prove with reasoning that it is impossible to find a value of $n$ that allows both overall limits to exist simultaneously, interpreting this fact using the one-sided limits obtained.