Application of the Derivative to Computing 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 $0$:

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

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

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

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:

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

Given the functions $f(x) = e^x - x - 1$ and $g(x) = x^2$, calculate with full reasoning the limit of their quotient as $x$ tends to $0$:

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

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

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?

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

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

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 the expression in the numerator must vanish at $x=0$ in order 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:

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

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 $b$.

It is known that the concentration of a chemical reactant in a limiting catalytic process follows dynamics such that:

$$ \lim_{x \to 0} \frac{\sin(ax) - bx}{x^3} = -\frac{4}{3} $$

Apply L'Hôpital's rule repeatedly to determine the exact numerical values of the real parameters $a$ and $b$.

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:

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

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

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

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

What value does the number of shares tend to at the initial moment? To find out, calculate:

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

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

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

The depreciation factor of a long-term financial asset is modelled by evaluating the following limit:

$$ L = \lim_{x \to \infty} \left( \frac{x^2 + 3x}{x^2 - x} \right)^x $$

Transform this indeterminate form of type $1^\infty$ using logarithms and apply L'Hôpital's rule to calculate the exact value of the depreciation factor $L$.

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

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

In a mathematical model of light diffraction through a microscopic slit, the following limit appears:

$$ \lim_{x \to 0} \frac{e^{2x} - 1 - 2x}{x^2} $$

Apply L'Hôpital's rule successively to determine its exact value.

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 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.

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.

A telecommunications engineer analyses the power loss of a signal in an optical fiber. The relative power $P$ with respect to distance $d$ is modelled using a function that requires resolving the following indeterminate form as 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, stating the initial indeterminate form and the algebraic transformations performed to apply L'Hôpital's Rule.

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.

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.

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

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

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

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.