L'Hôpital's Rule

Calculate the following limit using L'Hôpital's Rule, first verifying that the necessary indeterminate form is present:

$$\lim_{x \to 2} \frac{x^3 - 5x + 2}{x^2 - 4}$$

In the study of the propagation of an electromagnetic signal, the following expression appears to calculate the intensity at a critical point. Find the value of the limit as the time $t$ approaches $0$:

$$\lim_{t \to 0} \frac{e^{3t} - 1}{\sin(2t)}$$

Determine the value of the following limit of a logarithmic function. Remember to differentiate the numerator and the denominator correctly and independently:

$$\lim_{x \to 1} \frac{\ln(x)}{x^2 + 2x - 3}$$

Observe the following graph representing two functions $f(x)$ (in white) and $g(x)$ (in dark gray) that are zero at the point $x = a$. Knowing that $f'(a) = 6$ and $g'(a) = 2$, calculate the value of the limit:

$$\lim_{x \to a} \frac{f(x)}{g(x)}$$

Find the value of the parameter $k$ so that the following limit is equal to $4$:

$$\lim_{x \to 0} \frac{\tan(kx)}{x} = 4$$

It is known that the concentration of a chemical reactant in a mixture follows the function $C(t)$. Calculate the limiting concentration as the time $t$ approaches $0$ from the right:

$$\lim_{t \to 0^+} \frac{1 - \cos(t)}{t^2}$$

Given the functions $f(x) = e^x - x - 1$ and $g(x) = x^2$, find the limit of their quotient as $x$ approaches $0$:

$$\lim_{x \to 0} \frac{f(x)}{g(x)}$$

Note: You may need to apply L'Hôpital's Rule more than once.

In the study of the propagation of a Wi-Fi signal in an environment with interference, the loss of intensity has been modeled by the following function of the distance $x$ (in feet):

$$f(x) = \frac{e^{2x} - e^{-2x} - 4x}{x - \sin(x)}$$

Calculate the limit of this function as the distance $x$ approaches $0$, that is, $\lim_{x \to 0} f(x)$, to determine the behavior of the signal at the point of emission.

Calculate the limit of the following rational function at infinity. Although you could solve it by comparing degrees, use L'Hôpital's Rule to prove the result:

$$\lim_{x \to +\infty} \frac{5x^2 - 3x + 1}{2x^2 + 7}$$

It is desired to study the convergence of an artificial intelligence algorithm. The error rate as a function of the number of iterations $x$ is defined as:

$$E(x) = \frac{\ln(x^4 + 1)}{x}$$

Prove, using L'Hôpital's Rule, that as the number of iterations approaches infinity ($x \to \infty$), the error rate approaches $0$. What does this tell us about the long-term efficiency of the algorithm?

In a biotechnology laboratory, the growth of a bacteria population in a controlled environment follows a law whose cumulative accumulation rate is expressed by an integral. It is desired to study the behavior of the population density at the initial instant by calculating the following limit:

$$\lim_{x \to 0} \frac{\int_{0}^{x} \ln(1 + t^2) \, dt}{x^3}$$

Use L'Hôpital's Rule and the Fundamental Theorem of Calculus to determine the value of this limit.

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

$$f(x) = \frac{e^x - kx - 1}{x^2}$$

Calculate the value of $k$ so that the limit $\lim_{x \to 0} f(x)$ is a finite real number and determine this value. Explain why it is necessary for the expression in the numerator to be zero at $x=0$ to be able to apply the rule.

A team of sound engineers is analyzing the response of an audio filter. The gain of the filter as a function of the frequency $x$ is given by the expression:

$$G(x) = \frac{a \cdot \cos(x) + b \cdot e^x - 2}{x^2}$$

Knowing that the limit of the gain as the frequency approaches zero is equal to $1$, that is, $\lim_{x \to 0} G(x) = 1$, find the values of the parameters $a$ and $b$.

In the design of a new roller coaster, the curvature of an initial section is determined by the existence of the limit of a transition function. For the design to be viable, the following limit is required to be a finite real number:

$$\lim_{x \to 0} \frac{a \cdot \cos(x) + b \cdot e^x - 3 + x}{x^2}$$

Find the values of the parameters $a, b \in \mathbb{R}$ so that this limit exists and is finite. Once obtained, calculate the value of this limit.

Calculate the value of the following limit by first transforming the indeterminate form of type $\infty - \infty$ into a quotient to be able to apply L'Hôpital's Rule:

$$\lim_{x \to 1} \left( \frac{x}{x-1} - \frac{1}{\ln(x)} \right)$$

Calculate the limit of the following function as $x$ approaches $\frac{\pi}{2}$. Note that you will need to perform a prior transformation to convert the indeterminate form $0 \cdot \infty$ into a form suitable for L'Hôpital's Rule:

$$\lim_{x \to \frac{\pi}{2}} \left( \sin(x) - 1 \right) \cdot \tan(x)$$

A telecommunications engineer analyzes the power loss of a signal in an optical fiber. The relative power $P$ with respect to the distance $d$ is modeled by a function that requires solving the following indeterminate form when the signal approaches a critical node ($d \to 1$):

$$\lim_{d \to 1} \left( \frac{d}{\ln(d)} - \frac{1}{d-1} \right)$$

Calculate the value of the limit, indicating the initial type of indeterminate form and the algebraic transformations carried out to apply L'Hôpital's Rule.

On a social network, the number of shares of a viral video during the first few seconds has been modeled by the function $S(t) = (1+2t)^{1/t}$, where $t$ is the time in seconds.

What value does the number of shares approach at the initial instant? To do this, calculate:

$$\lim_{t \to 0} (1+2t)^{\frac{1}{t}}$$

Determine the value of the following limit which presents an indeterminate form of the type $1^\infty$. To solve it, use the properties of logarithms and L'Hôpital's Rule:

$$\lim_{x \to 0} (\cos x)^{\frac{1}{x^2}}$$

Calculate the value of the following limit, which presents an indeterminate form of the type $1^\infty$. Make sure to transform the expression appropriately before applying L'Hôpital's Rule:

$$\lim_{x \to 0} \left( \frac{\sin(x)}{x} \right)^{\frac{1}{x^2}}$$

Calculate the value of the limit of the following function as $x$ approaches zero from the right ($x \to 0^+$). Identify the type of indeterminate form and use the relationship between powers and exponentials ($u^v = e^{v \cdot \ln(u)}$):

$$\lim_{x \to 0^+} (\tan(x))^x$$

Determine, if they exist, the values of the parameters $a$ and $b$ so that the following function is continuous throughout its domain:

$$f(x) = \begin{cases} \dfrac{\sin(ax^2)}{x^2} & \text{if } x < 0 \\ b + 2 & \text{if } x = 0 \\ \dfrac{e^x - e^{-x} - 2x}{x - \sin(x)} & \text{if } x > 0 \end{cases}$$

To solve the right-hand limit, it will be necessary to apply L'Hôpital's Rule repeatedly. Justify each step of the process.

Consider the function $g(x) = x^2 \cdot e^{-1/x^2}$ for $x \neq 0$.

1. Study whether it is possible to assign a value to $g(0)$ so that the function is continuous at $x = 0$.

2. Using the definition of the derivative as a limit and applying L'Hôpital's Rule, prove whether the function is differentiable at $x = 0$.

Note: Take special care with the change of variable needed to solve the limit at infinity if necessary.

Given the function $f(x) = \frac{x^2 + kx + 2}{x^2 - 1}$, it is known that it has a removable discontinuity at $x = 1$.

1. Calculate the value of the constant $k$.

2. For this value of $k$, calculate the value the function should take at $f(1)$ for it to be continuous at that point, by solving the limit:

$$\lim_{x \to 1} \frac{x^2 + kx + 2}{x^2 - 1}$$

1. Is there any slant asymptote for this function? Justify your answer using limits.