Application to the calculation of limits. L'Hôpital's rule

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

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

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

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

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

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

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:

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

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

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

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

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

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

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

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, $\lim_{x \to 0} G(x) = -1$, calculate with reasoning 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 feasible, the following limit must be a finite real number:

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

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

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

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

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

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

In a biotechnology laboratory, the growth of a bacterial population in a controlled environment follows a law whose cumulative accumulation rate is expressed by an integral. We wish to study the behaviour 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.