Exercises: Continuity of a function at a point

Observe the following graph of the function $f(x)$ and analyze its behavior at the point with abscissa $x = 2$. Indicate whether the function is continuous at that point by checking the three conditions for continuity: existence of $f(2)$, existence of the $\lim_{x \to 2} f(x)$ and the coincidence of both values.

Study the continuity of the function $f(x) = \frac{x^2 - 1}{x - 1}$ at the point $x = 1$. If it presents a discontinuity, indicate what type it is (removable, finite jump or infinite jump) and reason whether there is any way to redefine the function so that it is continuous at that point.

Given the following piecewise function, calculate the one-sided limits at the breakpoint $x = 3$ and determine whether the function is continuous at that point:

$f(x) = \begin{cases} 2x - 4 & \text{if } x < 3 \\ \sqrt{x + 1} & \text{if } x \ge 3 \end{cases}$

A surveillance drone follows a height trajectory defined by the function $h(t)$ (in feet) with respect to time $t$ (in seconds). Due to a change in the control system, the function changes at $t = 5$:

$h(t) = \begin{cases} t^2 - 10 & \text{if } 0 \le t < 5 \\ 3t & \text{if } t \ge 5 \end{cases}$

Check if the height transition is continuous at the instant $t = 5$ to prevent the drone from suffering a sudden movement.

Determine the value of the parameter $k$ so that the function $g(x)$ is continuous over the entire real line $\mathbb{R}$:

$g(x) = \begin{cases} x + k & \text{if } x \le 2 \\ x^2 - 1 & \text{if } x > 2 \end{cases}$

Visually analyze the continuity of the function represented in the following graph. Identify the value of $x$ where the function presents a discontinuity and classify it according to the behavior of its lateral branches.

Given the function $f(x) = \frac{1}{x + 4}$, justify why it is not continuous at the point $x = -4$. To do this, calculate the one-sided limits $\lim_{x \to -4^-} f(x)$ and $\lim_{x \to -4^+} f(x)$ and describe the type of discontinuity it presents.

Check if the function $f(x) = e^{x-1}$ is continuous at the point $x = 1$. For the answer to be complete, you must verify whether the value of the function at the point coincides with the limit of the function as $x$ approaches 1.

Given the function dependent on the parameter $k \in \mathbb{R}$:

Calculate the value of $k$ so that the function presents a removable discontinuity at $x = 3$. Is there any value of $k$ that makes it continuous at that point? Reason your answer by calculating one-sided limits.

Study the continuity of the following function over its entire domain, classifying the points of discontinuity you find:

Accompany your analytical analysis with a sketch of the one-sided limits at the conflicting points.

An industrial cooling system regulates its power $P$ (in kilowatts) according to the outside temperature $T$ (in degrees Fahrenheit) using the following function:

Determine the value of the constant $a$ so that the power transition is continuous when the temperature passes through $0^\circ C$.

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

Find the values of $a$ and $b$ so that the function is continuous in its domain of definition. Once found, indicate whether the function is continuous at $x = -1$.

Classify all the discontinuities of the function $g(x)$ and determine its vertical asymptotes by analyzing the behavior of the function in the vicinity of the values where it is not defined:

How would you define the function at the points of removable discontinuity so that it becomes continuous at them?

A new engine prototype is being tested whose acceleration $a$ (in $m/s^2$) depends on time $t$ (in seconds) according to the expression:

Calculate the value of $m$ so that the acceleration does not suffer sudden jumps (is continuous) at the instant $t = 2$ s.

Given the function:

Study its continuity at the points $x = 1$ and $x = -1$. If there are discontinuities, indicate what type they are and if there is any way to redefine the function to eliminate them.

Determine the values of $a$ and $b$ so that the function $f(x)$ is continuous over all $\mathbb{R}$:

Explain in detail the process followed, verifying the equality of the one-sided limits at each branch change point.

In the design of a new component for a particle accelerator, the intensity of a magnetic field as a function of the position $x$ (in inches) is defined by the following function:

$f(x) = \begin{cases} \frac{e^{ax} - 1}{x} & \text{if } x < 0 \\ b & \text{if } x = 0 \\ \frac{x^2 - 4}{x - 2} \cdot \cos(\pi x) & \text{if } x > 0 \end{cases}$

Determine the values of the real parameters $a$ and $b$ so that the intensity of the magnetic field is continuous at the point $x = 0$. Once found, would the function be continuous at the point $x = 2$? Reason your answer analytically.

Study the continuity of the following function over its entire domain, classifying each of its points of discontinuity in detail:

$f(x) = \frac{|x^2 - 1|}{x^2 - 3x + 2}$

Subsequently, propose a redefinition for the function at those points where the discontinuity is removable to transform it into a continuous function at those points.

A telecommunications company is adjusting the power of a 5G antenna. The signal loss $L$ (in decibels) depends on the distance $d$ (in miles) according to the expression:

$L(d) = \begin{cases} \frac{k \cdot \ln(d+1)}{d} & \text{if } 0 < d < 1 \\ \sqrt{d^2 + 3} - a & \text{if } d \geq 1 \end{cases}$

The power transition is required to be smooth and continuous so as not to interrupt calls at the 1 mile limit. Calculate the values of $k$ and $a$ knowing that as the distance tends to 0 miles, the signal loss tends to 2 dB.

Given the function $f(x) = \left( \frac{2x^2 + 3}{2x^2 - 1} \right)^{ax^2}$:

1. a) Find the value of the parameter $a \in \mathbb{R}$ so that $\lim_{x \to \infty} f(x) = e^2$.
2. b) With the value of $a$ obtained in the previous section, define a new function $g(x)$ such that:

$g(x) = \begin{cases} f(x) & \text{if } x > 1 \\ \frac{x^2 + bx + c}{x - 1} & \text{if } x < 1 \end{cases}$

Determine $b$ and $c$ so that $g(x)$ is continuous at $x = 1$.

Apply Bolzano's Theorem to prove that the graphs of the functions $f(x) = \ln(x)$ and $g(x) = e^{-x}$ intersect at at least one point within the interval $(1, e)$. Justify with reasoning whether this point of intersection is unique in that interval based on the growth of the functions.

Study the continuity of the function $f(x) = \frac{1}{1 + 2^{1/x}}$ at the point $x = 0$. To do this, calculate the one-sided limits $\lim_{x \to 0^-} f(x)$ and $\lim_{x \to 0^+} f(x)$. What type of discontinuity does the function present at this point?

A team of robotics engineers has programmed the movement of a mechanical arm. The velocity of the end of the arm (in ft/s) follows the trajectory:

$v(t) = \begin{cases} \frac{\sqrt{t+5}-3}{t-4} & \text{if } 0 \leq t < 4 \\ \frac{a}{12} & \text{if } t = 4 \\ \frac{t^2 - 16}{t^2 - 3t - 4} \cdot b & \text{if } t > 4 \end{cases}$

Determine the values of $a$ and $b$ so that the robot's movement is continuous (without sudden jerks) at the instant $t = 4$ seconds.

Given the function dependent on the parameter $k$:

$f(x) = \frac{x^3 - kx^2 + 3x}{x^2 - x}$

1. a) Find the value of $k$ so that the function has a removable discontinuity at $x = 1$.
2. b) For that value of $k$, study the continuity at $x = 0$ and determine if there is any vertical asymptote.
3. c) Is there any value of $k$ that makes the function continuous over all $\mathbb{R}$? Justify your answer.