Exercises: Continuity

Observe the following graph of the function $f(x)$ and determine the value of the indicated limits. If the limit does not exist or is infinite, specify so.

1. a) $\displaystyle \lim_{x \to -2} f(x)$
2. b) $\displaystyle \lim_{x \to 0} f(x)$
3. c) $\displaystyle \lim_{x \to 3^-} f(x)$
4. d) $\displaystyle \lim_{x \to 3^+} f(x)$

Calculate the following limits of polynomial and rational functions using direct substitution:

1. a) $\displaystyle \lim_{x \to 2} (x^2 - 5x + 6)$

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

1. c) $\displaystyle \lim_{x \to 0} \frac{e^x + 2}{x + 4}$

A study on the temperature of an automated greenhouse indicates that it follows the function $T(t) = \frac{20t + 100}{t + 2}$, where $t$ is the time in hours since opening and $T$ is the temperature in °C.

1. a) What temperature does the greenhouse tend to after a long time ($t \to +\infty$)? Calculate the limit.

1. b) Illustrate the concept with a sketch of the graph of $T(t)$ for $t \geq 0$, showing the horizontal asymptote. Calculate $T(0)$ and some other value to place the curve properly.

Given the function $f(x) = \frac{x + 3}{x - 4}$, calculate the one-sided limits as $x$ approaches $4$ to determine whether the limit at that point exists:

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

1. b) $\displaystyle \lim_{x \to 4^+} \frac{x + 3}{x - 4}$

Analyse the continuity of the following piecewise-defined function at the split point $x = 2$. To do this, check whether the three conditions for continuity are satisfied:

$f(x) = \begin{cases} 2x + 1 & \text{if } x < 2 \\ 7 - x & \text{if } x \geq 2 \end{cases}$

Match each of the following graphical situations with its corresponding type of discontinuity: removable, finite jump, or infinite jump.

Determine the value of the parameter $k$ so that the function $g(x)$ is continuous on the set of all real numbers:

$g(x) = \begin{cases} kx^2 - 1 & \text{if } x \leq 3 \\ 4x + 5 & \text{if } x > 3 \end{cases}$

A courier company charges $5$ € for shipments weighing up to $2$ kg (including $2$ kg). For weights greater than $2$ kg, the rate is given by the function $P(w) = 2w + 1$, where $w$ is the weight in kg and $P(w)$ is the cost in euros.

1. a) Express the cost function $C(w)$, for $w > 0$, as a piecewise-defined function.

1. b) Is the cost function continuous at the point $w = 2$? Justify your answer by calculating the one-sided limits.

Lucas is studying the continuity of the function $f(x) = \frac{x^2 - 9}{x - 3}$ at the point $x = 3$. On his exam, he writes the following:

"I factor and simplify the fraction: $\frac{(x-3)(x+3)}{x-3} = x+3$. Since substituting $x=3$ yields $3+3=6$, the function is continuous at $x=3$ and its value is $f(3)=6$."

1. a) Explain with mathematical rigor what conceptual error Lucas has made regarding the formal definition of continuity.
2. b) Classify the type of discontinuity that $f(x)$ actually has at $x = 3$.
3. c) Explain how this discontinuity could be "repaired" by defining a piecewise function so that it is continuous on all of $\mathbb{R}$.

In an economic study on the price elasticity of demand for a tech product, an analyst models the variation in marginal profit using the function $I(x) = \frac{\sqrt{x+9}-3}{x}$, where $x$ represents the deviation in advertising spending in thousands of euros from the baseline budget ($x=0$).

1. a) Estimate: Before performing algebraic operations, approximate the value of $\lim_{x \to 0} I(x)$ by calculating the numerical value of $I(x)$ for $x = -0,1$; $x = -0,01$; $x = 0,01$ and $x = 0,1$. What value does the function seem to approach?
2. b) Check: Calculate the exact value of $\lim_{x \to 0} I(x)$ analytically by multiplying by the conjugate expression and check whether your initial estimate was correct.
3. c) Knowing that $I(0)$ is not originally defined in the formula, classify the discontinuity that $I(x)$ has at $x = 0$ and state what exact value should be assigned to $I(0)$ for the model to be continuous.

The annual maintenance cost of a server hosting a social network depends on the number of terabytes ($x$) of data processed, according to the function:

$$C(x) = \frac{12x^2 + 500x + 1000}{0,5x^2 + 10x}$$

Where $C(x)$ is the average annual maintenance cost per terabyte, expressed in euros. Engineers want to know what the average maintenance cost per terabyte will be when the volume of data processed is extremely large. Calculate the limit of the function as $x \to +\infty$ and interpret the result in the context of the problem.

A financial institution offers a savings account whose return is given by the expression $M(n) = \left(1 + \frac{0,04}{n}\right)^{2n}$, where $n$ represents the number of times interest is compounded per year. An economics student wants to investigate what happens to the final capital if compounding were instantaneous, that is, if the number of compounding periods tended to infinity.

Calculate the value of $\lim_{n \to \infty} M(n)$ using the relationship with the number $e$.

The profile of a ramp for a skate park is being designed. The design combines a curved (parabolic) section with a straight section to ensure a smooth transition. The height $h$ in metres with respect to horizontal distance $x$ is given by:

$$h(x) = \begin{cases} \frac{x^2}{4} + 1 & \text{if } x \le 2 \\ ax + b & \text{if } x > 2 \end{cases}$$

Knowing that for the ramp not to be dangerous the function must be continuous at $x = 2$ and that the gradient of the line must be $1$, determine the values of $a$ and $b$.

Given the following graph of a function $f(x)$, analyse its behaviour at the indicated points by calculating its one-sided limits and determining whether the overall limit exists. Also classify the type of discontinuity it has in each case:

1. At $x = -2$

2. At $x = 1$

3. At $x = 3$

A firework rocket ascends during the town festival. Its height in metres $H$ after $t$ seconds of flight is defined by the function:

$$H(t) = \frac{t^2 - 16}{t - 4}$$

1. Calculate the value of the height as $t$ approaches $4$ seconds.

2. Is the height defined at the exact moment $t = 4$? Justify whether the function is continuous at that point.

3. If it is not continuous, redefine the function so that the flight is "smooth" (continuous) at that moment.

Calculate the value of the following limits by resolving any indeterminate forms that appear:

1. a) $\lim_{x \to 3} \frac{\sqrt{x+1} - 2}{x - 3}$

1. b) $\lim_{x \to 2} \frac{x^2 - 5x + 6}{x^2 - 4}$

In a study on the growth of a lynx population in Doñana, biologists have modelled the number of individuals using the function:

$$P(t) = \frac{250t + 120}{0,5t + 2}$$

where $t$ is the time in years since the start of the study.

1. How many lynxes were there at the start of the study ($t=0$)?

2. What value does the lynx population tend to in the long run? Calculate the limit as $t \to +\infty$.

3. Find the equations of the asymptotes of this function.

Determine the value of the parameters $k$ and $m$ so that the following function is continuous throughout its domain ($\mathbb{R}$):

$$f(x) = \begin{cases} kx - 2 & \text{if } x < -1 \\ x^2 + x & \text{if } -1 \le x < 2 \\ m & \text{if } x \ge 2 \end{cases}$$

Once the values have been found, explain what conditions had to be satisfied at the split points $x = -1$ and $x = 2$ to guarantee continuity.

To encourage water conservation in a municipality, the municipal company establishes a bimonthly rate based on the consumed volume $x$ (in $\text{m}^3$). The bill amount $C(x)$ (in euros) is given by the following piecewise function:

$$C(x) = \begin{cases} 15 + 0,80x & \text{if } 0 \le x \le 20 \\ a + 1,40(x - 20) & \text{if } 20 < x \le 50 \\ b + 2,10(x - 50) & \text{if } x > 50 \end{cases}$$

1. a) Municipal regulations require the bill to be a continuous function of consumption so as not to arbitrarily penalize users at the bracket transition points. Determine the values of $a$ and $b$ so that $C(x)$ is continuous across its entire domain ($x \ge 0$).
2. b) Interpret the economic meaning of the constants $a$ and $b$ obtained within the context of the rate structure.
3. c) If the company had mistakenly set $a = 35$, calculate the one-sided limits at $x = 20$, classify the resulting discontinuity, and indicate how many euros more a user consuming $20,01\text{ m}^3$ would unfairly pay compared to one consuming exactly $20\text{ m}^3$.

Design the algebraic expression of a rational function of the form $f(x) = \frac{P(x)}{Q(x)}$ (where $P(x)$ and $Q(x)$ are degree 2 polynomials with real coefficients) that simultaneously satisfies the following three conditions:

1. It has a removable discontinuity at the point $x = 1$, satisfying $\lim_{x \to 1} f(x) = 3$.

2. It has an infinite discontinuity (vertical asymptote) at the line $x = -2$.

3. It has a horizontal asymptote at the line $y = 1$ as $x \to +\infty$.

Justify in detail how you determine the roots and coefficients of $P(x)$ and $Q(x)$ from each of the given conditions, write the simplified function, and analytically verify that the three requirements are met by calculating the corresponding limits.

The evolution of the profits of a fintech company during its first years of existence is being studied. The function that models the accumulated profit $B(t)$ (in millions of euros) as a function of time $t$ (in years) is given by the following expression:

$$B(t) = \begin{cases} \displaystyle \frac{at^2 + bt}{t^2 + 1} & \text{if } 0 \le t \le 2 \\ \displaystyle \frac{t-2}{\sqrt{t+7}-3} & \text{if } t > 2 \end{cases}$$

Calculate the values of $a$ and $b$ so that the profit is a continuous function on $t = 2$, knowing that in the first year ($t = 1$) the company's profit was exactly $2$ million euros.

Given the function $f(x) = \displaystyle \frac{2x^3 - 5x^2 - 4x + 3}{x^2 - x - 6}$, carry out an exhaustive study of its asymptotes. To do this, you must:

1. Find the vertical asymptotes and study the behavior of the function on both sides of them using one-sided limits.

2. Check whether there is a horizontal or oblique asymptote.

3. Determine the relative position of the curve with respect to the non-vertical asymptote found.

A financial analyst uses a mathematical model to predict the value of an aggressive investment portfolio. The return $R(x)$ depends on a market index $x$, but the model has a discontinuity at a critical value:

$$R(x) = \begin{cases} \displaystyle \frac{|x-3|}{x-3} + kx & \text{if } x < 3 \\ \displaystyle \frac{x^2 - 9}{x^2 - 5x + 6} & \text{if } x > 3 \end{cases}$$

1. Calculate the value of $k$ so that the limit of the function exists as $x$ tends to $3$.

2. Once $k$ has been found, is the function continuous at $x = 3$? Justify your answer by analysing the types of discontinuity.

In a digital sociology laboratory, it has been observed that the number of users (in thousands) who share a fake news story follows the model $U(t) = \left( \displaystyle \frac{3t^2 + 5}{3t^2 + 2} \right)^{2t^2+1}$, where $t$ is the time in hours since publication.

Determine the number of users that the news story will tend to in the "long run" (as time tends to infinity). Explain with reasons the type of indeterminate form that appears and the process to solve it.

Find the values of the parameters $a$ and $b$ so that the following function is continuous throughout its domain:

$$f(x) = \begin{cases} \displaystyle \frac{\sqrt{x+3}-2}{x-1} & \text{if } x < 1 \\ a \cdot \ln(x) + b & \text{if } 1 \le x \le e \\ \displaystyle \frac{x-e}{x^2 - e^2} & \text{if } x > e \end{cases}$$

Note: Remember that $\ln(e) = 1$ and pay attention to indeterminate forms at the split points.

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

1. For which values of $m$ does the function have a removable discontinuity at $x = 2$ or $x = -2$?

2. For $m = -2$, classify all the discontinuities of the function and find its asymptotes.

3. Draw a sketch of the infinite branches for the case $m = -2$.

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

An investment fund guarantees monthly management fees calculated by the function $G(x) = \sqrt{x^2 + 4x} - x$, where $x$ represents the invested capital in thousands of euros. The graph of the function is as follows:

1. a) Calculate the limit of the fees as the investment $x$ grows indefinitely, that is, $\lim_{x\to+\infty} G(x)$. Keep in mind that an $\infty-\infty$ indeterminate form appears.

1. b) If a client invests a very small amount (tending to zero from the right), what value do their management fees approach? Calculate $\lim_{x\to 0^+} G(x)$.

1. c) Check that your results are consistent with the graph and interpret the result of part a) from the point of view of investment efficiency for large capitals.

Given the function $h(x) = \displaystyle \frac{x^2 - kx + 4}{x^3 - 4x}$, determine the possible values of $k$ knowing that the function has a removable discontinuity. For each value of $k$ obtained:

1. Identify the point where the discontinuity is removable.

2. Calculate the one-sided limits at the remaining points where the function is not continuous.

3. Determine all the asymptotes of the resulting function.

We want to study the value of the parameter $b$ that guarantees the continuity at $x = 0$ of the function:

$$f(x) = \begin{cases} (1 + 3x)^{\frac{2}{x}} & \text{if } x > 0 \\ a \cdot x + b & \text{if } x \le 0 \end{cases}$$

To calculate the right-hand limit $\lim_{x \to 0^+} (1+3x)^{\frac{2}{x}}$, three students propose different strategies:

• Method 1 (Sofía): Algebraically manipulates the base and exponent to directly apply the classical definition of Euler's number $e$, transforming the expression into $\left[(1+3x)^{\frac{1}{3x}}\right]^6$.

• Method 2 (Marcos): Applies the general formula for indeterminacies of the type $1^\infty$: $\lim g(x)^{h(x)} = e^L$, where $L = \lim_{x \to 0^+} h(x) \cdot (g(x) - 1)$.

• Method 3 (Elena): Substitutes decreasing very small values of $x$ such as $x = 0,001$ and $x = 0,0001$ with the calculator and rounds the numerical result.

1. a) Evaluate the formal validity of the three methods. Explain why Elena's method does not constitute a valid mathematical proof of continuity despite providing an intuitive approximation.
2. b) Compare Sofía's and Marcos's methods: which one is more direct in this specific exercise, and which one has a lower risk of error if the base were a complex fraction like $\frac{2x+3}{x+3}$? Justify your answer.
3. c) Choose one of the two rigorous methods, calculate the right-hand limit in detail, and deduce the exact value of $b$ for $f(x)$ to be continuous at $x = 0$. Does the value of parameter $a$ influence the continuity at the point $x = 0$? Reason why.

In designing the income tax of an autonomous community, two proposals are debated to calculate the tax liability $C(x)$ (in thousands of euros) based on the taxable base $x$ (in thousands of euros), with a tax rate change when exceeding $20$ thousand euros ($15\,\%$ in the first bracket and $30\,\%$ from then on):

• Proposal A (flat rate applied to the total based on the bracket reached):

$$C_A(x) = \begin{cases} 0,15x & \text{if } 0 \le x \le 20 \\ 0,30x & \text{if } x > 20 \end{cases}$$

• Proposal B (progressive marginal brackets):

$$C_B(x) = \begin{cases} 0,15x & \text{if } 0 \le x \le 20 \\ 0,15 \cdot 20 + 0,30(x - 20) & \text{if } x > 20 \end{cases}$$

1. a) Analytically study the continuity of $C_A(x)$ and $C_B(x)$ at the critical point $x = 20$. If either has a discontinuity, classify it and calculate the jump size.
2. b) The taxpayer's disposable net income after paying the tax is defined as $S(x) = x - C(x)$. Write the piecewise analytical expression for $S_A(x)$ and $S_B(x)$, and analyze the continuity of both at $x = 20$.
3. c) Calculate the net income left for a taxpayer with a base of $20$ thousand euros and for another with a base of $20,01$ thousand euros (just ten euros more in gross income) under both systems. Use the concepts of limit and continuity to explain what economic anomaly Proposal A generates (known in economics as the "cliff effect" or jump error) and why the mathematical continuity of Proposal B is essential to ensure tax fairness.