Exercises: Application to finding 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:

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

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 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 $g(x) = x^2$, 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.

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:

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

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.

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:

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

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 $b$.

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:

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

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?

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.

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 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.

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.

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:

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

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.

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 ($u^v = e^{v \times \ln(u)}$):

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

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.

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 $b$.

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 $L$.