Exercises: Continuity of some functions

Determine the points of discontinuity of the rational function $f(x) = \frac{2x+5}{x^2-9}$ and indicate, by calculating the one-sided limits, the behavior of the function at these points.

Find the domain of continuity of the radical function $g(t) = \sqrt{t^2-4t}$. Justify your answer by solving the corresponding inequality.

Study the continuity of the following piecewise function at the point $x=1$:

$h(x) = \begin{cases} 3x-1 & \text{if } x < 1 \\ x^2+1 & \text{if } x \ge 1 \end{cases}$

Calculate the value of the parameter $k$ so that the function $f(x)$ is continuous over its entire domain:

$f(x) = \begin{cases} x^2+k & \text{if } x \le 2 \\ 5x-3 & \text{if } x > 2 \end{cases}$

Observe the following graph representing the water level in a tank $N(t)$ as a function of time $t$.

Identify the type of discontinuity the function presents at the instant $t = a$ and express the one-sided limits mathematically using $L_1$ and $L_2$.

Given the logarithmic function $f(x) = \ln(8-2x)$, determine its domain of continuity, justifying the necessary calculations.

Study the continuity of the trigonometric function $g(x) = \frac{1}{\cos x}$ in the open interval $(0, 2\pi)$. At which points does it present discontinuities and of what type are they?

A parking rate charges $\$2$ for the first hour or fraction thereof, and $\$3.5$ if the time exceeds the first hour up to a maximum of two hours. Define the cost function $C(t)$ for $t \in (0, 2]$ and analyze whether it is continuous at $t=1$.

Find the values of the parameters $a$ and $b$ so that the following function is continuous over its entire domain $\mathbb{R}$:

$f(x) = \begin{cases} ax^2+b & \text{if } x < 1 \\ 3x-a & \text{if } 1 \le x < 3 \\ \frac{b}{x-2} & \text{if } x \ge 3 \end{cases}$

In a laboratory culture, the number of bacteria $N(t)$ (in thousands) as a function of time $t$ (in hours) is modeled by the following function:

$N(t) = \begin{cases} 10 e^{0.5t} & \text{if } 0 \le t < 4 \\ a t + b & \text{if } t \ge 4 \end{cases}$

Knowing that the model must be continuous at $t=4$ and that at $6$ hours there are exactly $100$ thousand bacteria, determine the values of $a$ and $b$.

Determine the value of the parameter $k$ so that the rational function $f(x) = \frac{x^2-kx-6}{x-2}$ has a removable discontinuity at $x=2$. Once $k$ is found, what value should $f(2)$ take for the function to be completely continuous at that point?

Given the absolute value function $f(x) = \frac{x^2-9}{|x-3|}$, rewrite it as a piecewise function and classify the type of discontinuity it presents at $x=3$.

The voltage $V(t)$ in volts of a closed electrical circuit behaves according to the function:

$V(t) = \begin{cases} 12\cos(\pi t) & \text{if } t \le 1 \\ k e^{-(t-1)} & \text{if } t > 1 \end{cases}$

Calculate the value of $k$ so that the voltage does not experience a sudden jump (is continuous) at the instant $t=1$.

Study the points of discontinuity of the function $f(x) = \frac{2x^2-8}{x^2-x-2}$. Classify each of them, indicating whether they are removable or non-removable (and of what type).

The following graph illustrates the volume $V(t)$ in gallons of an industrial water tank during an emptying and subsequent filtering process, as a function of time $t$ in hours.

Knowing that for $t > 5$ the curve follows an exponential model $V(t) = A \cdot e^{-k(t-5)}$, and assuming that the function is continuous over its entire domain, determine the value of $A$.

Analyze the continuity of the function $f(x) = \frac{x}{1-e^{1/x}}$ at the point $x=0$. Calculate the one-sided limits and determine whether the discontinuity is removable or a jump discontinuity.

The following piecewise function is defined, including radicals, exponentials, and polynomials:

$f(x) = \begin{cases} \frac{\sqrt{x^2+a^2}-a}{x^2} & \text{if } x < 0 \\ b & \text{if } x = 0 \\ \frac{e^{cx}-1}{4x} & \text{if } x > 0 \end{cases}$

Knowing that $a > 0$, determine the relationship between the parameters $a, b$ and $c$ so that the function is continuous at $x=0$.

In the design of a support beam, the shear force $S(x)$ along its length $x$ is given by:

$S(x) = \frac{k \sqrt{x}}{x^2-L^2}$

where $L$ is the total length of the beam ($L > 0$) and $k$ is a material constant. Study the domain of continuity of $S(x)$ and mathematically classify the discontinuity at $x=L$. What physical interpretation does this limit have in the context of the material's resistance?

The concentration of a drug in the bloodstream, $C(t)$ in mg/L, is modeled by the following absorption and elimination function:

$C(t) = \begin{cases} \frac{A t}{t^2+4} & \text{if } 0 \le t \le 2 \\ B e^{-0.5t} & \text{if } t > 2 \end{cases}$

Knowing that the function is continuous and that at $t=2$ the maximum concentration of $5$ mg/L is reached, calculate the exact values of $A$ and $B$. What happens to the concentration in the long term ($\lim_{t \to \infty} C(t)$)?

A progressive tax system is defined by the function $T(x)$, which represents the tax to be paid for a taxable base of $x$ dollars:

$T(x) = \begin{cases} 0.15x & \text{if } 0 \le x \le 12000 \\ a + 0.25(x-12000) & \text{if } 12000 < x \le 30000 \\ b + 0.40(x-30000) & \text{if } x > 30000 \end{cases}$

Calculate $a$ and $b$ so that the tax function is continuous (i.e. there are no sudden jumps in tax payment when changing brackets).

The specific heat capacity $C_p(T)$ of a material undergoes a phase transition at a critical temperature $T_c = 100$ K. The proposed thermodynamic model is:

$C_p(T) = \begin{cases} \frac{500}{\sqrt{120-T}} & \text{if } T < 100 \\ 200 \ln(T-99) + k & \text{if } T \ge 100 \end{cases}$

Calculate the value of $k$ knowing that during the phase transition a finite jump discontinuity (latent heat) occurs such that $\lim_{T \to 100^+} C_p(T) - \lim_{T \to 100^-} C_p(T) = 80$.

Find the values of $a, b$ and $c$ so that the rational function $f(x) = \frac{ax^2+bx+c}{x^2-4}$ simultaneously satisfies the following conditions:

1. It has a removable discontinuity at $x=2$.

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

3. Its limit as $x \to \infty$ is $3$.

4. It crosses the y-axis at $y = -1.5$.

Study the continuity of the function $f(x) = e^{\frac{1}{|x|-1}}$ in detail. Determine its domain, calculate the limits at the critical points, and rigorously classify the type of discontinuity it presents at each of them.

The voltage signal on an oscilloscope shows a damped wave that suddenly stabilizes due to the action of a clipping diode at $t = t_c$.

The analytical model for $t \ge 0$ is:

$V(t) = \begin{cases} e^{-0.5t} \cos(2\pi t) & \text{if } 0 \le t < t_c \\ C & \text{if } t \ge t_c \end{cases}$

Knowing that the clipping occurs exactly at the fourth complete cycle of the cosine wave ($t_c = 4$), analytically prove that the function is continuous at that point by calculating the exact value that the constant $C$ must have.