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:

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

We wish to calculate the following limit at infinity:

$$\lim_{x \to +\infty} \frac{2x + 1}{\sqrt{4x^2 + 3}}$$

1. a) Verify that it presents an indeterminate form of type $\frac{\infty}{\infty}$ and attempt to evaluate it by directly applying L'Hôpital's rule, differentiating the numerator and the denominator. What happens upon successive differentiation? Explain why this method does not allow you to evaluate the limit.
2. b) Propose and develop an alternative algebraic method that allows calculating the limit in an immediate and rigorous way.
3. c) Based on the previous results, establish a general criterion to decide when it is advisable to prioritize algebraic simplification over the mechanical use of L'Hôpital's rule when dealing with quotients with radicals at infinity.

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

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

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

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

1. Does an oblique asymptote exist for this function? Justify your answer using limits.

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:

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

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:

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

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.

Determine, if they exist, the values of the parameters $a$ and $b$ so that the following function is continuous on its entire 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 find the limit from the right, it will be necessary to apply L'Hôpital's Rule repeatedly. Justify each step of the process.

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.

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

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

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

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