Continuity

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.

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 $b$.

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 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 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.

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.