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 $5$ pesos 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 pesos.

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.

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 pesos. 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 meters 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 slope of the line must be $1$, determine the values of $a$ and $b$.

Given the following graph of a function $f(x)$, analyse its behavior 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 meters $H$ 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.

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 pesos) 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 pesos.

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:

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

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 pesos. 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.