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 grey) that vanish at point $x = a$. Given that $f'(a) = 6$ and $g'(a) = 2$, calculate the value of the limit:

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

Mateo is calculating the following limit in a Mathematics II exam:

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

On his paper, he writes the following steps:

1) "Substituting $x = 0$, we get $\frac{e^0 - 0 - 1}{0^2 + 0} = \frac{0}{0}$. Since it is an indeterminate form, I apply L'Hôpital's rule by differentiating numerator and denominator: $\lim_{x \to 0} \frac{e^x - 1}{2x + 2}$".

2) "Since the variable $x$ still remains in the denominator, I apply L'Hôpital's rule again: $\lim_{x \to 0} \frac{e^x}{2} = \frac{e^0}{2} = \frac{1}{2}$".

Identify the conceptual error that Mateo made in step 2, explain in detail why it is not legitimate to apply the rule at that point, and calculate the correct value of the limit, justifying each step.

Consider the limit:

$$L = \lim_{x \to 0} \frac{e^{3x} - 1}{2x}$$

1. a) Before calculating it formally, estimate its numerical value: knowing that $e^{0,03} \approx 1,03045$ and that $e^{-0,03} \approx 0,97045$, evaluate the quotient at $x = 0,01$ and at $x = -0,01$. From these approximations, conjecture which exact decimal number represents $L$.
2. b) Check your conjecture by evaluating the limit using L'Hôpital's rule, verifying explicitly that the hypotheses required for its application are satisfied.
3. c) Compare the theoretical value obtained with your numerical estimates and explain why the mean of the values evaluated to the left and to the right is so close to the exact limit.

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.

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:

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

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 $b$.

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 $0 \cdot \infty$ into a form suitable for L'Hôpital's rule:

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

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.

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.

In the Spanish electricity market (OMIE), during an episode of strong wind power surge, the marginal clearing price $P(t)$ (in euros/MWh) as a function of the time $t$ elapsed since the start of the wind injection (in hours, for $t > 0$) is modeled by the expression:

$$P(t) = 45 + \frac{30 \cdot (\sin(2t) - 2t)}{t^3}$$

1. a) Analyze which indeterminate form the calculation of the instantaneous price at the moment of initial coupling leads to, that is, when calculating $\lim_{t \to 0^+} P(t)$.
2. b) Apply L'Hôpital's rule with justification as many times as necessary to calculate this limit, verifying the required conditions at each of the intermediate derivatives.
3. c) Interpret the result obtained in the context of the power system: does the price collapse to zero, tend to infinity, or stabilize at a positive threshold value at the beginning of the episode? Determine said limit price in euros/MWh.

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.

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.

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 fibre. 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 $g(x) = x^2 \cdot e^{-1/x^2}$ 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 ($u^v = e^{v \cdot \ln(u)}$):

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

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 $L$.

An engineering team is designing the smooth braking profile of a high-speed train in the Spanish railway network. Such behavior requires determining a family of functions that satisfies the curvature condition:

$$\lim_{x \to 0} \frac{a \cdot x + b \cdot x^2 + c \cdot x^3 + d \cdot \sin(x) - \sin(2x)}{x^3} = 4$$

where $a, b, c, d \in \mathbb{R}$ are design parameters.

1. a) Prove, by repeatedly applying L'Hôpital's rule and justifying the vanishing of the numerators at each step, the linear equations that the parameters must satisfy for the limit to exist and equal exactly $4$.
2. b) Reason whether a unique function with these properties exists or whether the solution has degrees of freedom. Determine the general solution of the system of parameters and write the specific analytical expression of two different functions that satisfy the required condition.
3. c) If the additional physical constraint that the linear coefficient of the intrinsic response vanishes ($a = 0$) is imposed, calculate the unique numerical values of $b$, $c$, and $d$.

Consider the function $f(x) = (a + b \cdot \cos x)^{1/x^2}$, defined on a punctured neighborhood of the origin ($x \ne 0$), with $a, b \in \mathbb{R}$.

1. a) Prove with reasoning that, for the limit $L = \lim_{x \to 0} f(x)$ to be a strictly positive and finite real number different from $1$, it is a necessary condition that $a + b = 1$. Justify in detail what would happen to said limit if $a + b > 1$ or if $0 < a + b < 1$.
2. b) Assuming that $a + b = 1$, transform the indeterminate form of type $1^\infty$ using natural logarithms and apply L'Hôpital's rule to deduce an expression for the value of the limit solely in terms of the parameter $a$.
3. c) Determine the exact values of $a$ and $b$ so that the limit is exactly $\frac{1}{\sqrt{e}}$. Next, investigate whether it is possible to find values of $a$ and $b$ such that the limit is $e^2$, verifying whether in that case the base function $a + b \cdot \cos x$ remains strictly positive in a neighborhood of $x = 0$.

Unit challenge. How does electricity demand evolve in our country, and what risk is there of grid overload? Become an energy analyst for a day.

1. Data and fitting: Go to the Red Eléctrica (REE) portal, download the hourly mainland demand data for a single day ($24\text{ h}$), and import them into GeoGebra or a spreadsheet to fit a differentiable polynomial function $D(t)$.

2. Differentiation and variation: Calculate the average rate of change during the period of highest morning consumption. Apply standard differentiation rules to analytically obtain the derivative function $D'(t)$, identifying the instantaneous rate and the moments of extreme consumption ($D'(t) = 0$).

3. L'Hôpital and probability: Evaluate the limiting sensitivity of the system to quadratic variations in time by solving $\lim_{t \to t_0} \frac{D(t) - D(t_0)}{t^2 - t_0^2}$ (with $t_0 \neq 0$) using L'Hôpital's rule. Furthermore, using the frequency table for the $24\text{ h}$ and the properties of probability (complementary event and union of compatible events), estimate the probability that a randomly chosen hour exhibits high demand (above the daily mean) or consumption acceleration ($D''(t) > 0$).

4. Report: Prepare a one-page report containing graphs, analytical derivations, and conclusions regarding the stability of the system.

Assessment criteria:

• Rigour in the analytical calculation of derivatives and the application of L'Hôpital's rule.

• Accuracy in fitting real data using GeoGebra and calculating rates of change.

• Correct use of probability properties and coherence in the report's conclusions.