Exercises: Calculation of limits. Indeterminate forms

To understand the concept of a limit, let us analyse the behaviour of the function $f(x) = \frac{x^2 - 4}{x - 2}$ as the values of $x$ get very close to $2$, without being exactly $2$.

Complete the following table of values using your calculator and predict what the value of $\displaystyle \lim_{x \to 2} f(x)$ will be:

Given the graph of the function $f(x)$ shown below, determine visually the value of the following limits:

1. a) $\displaystyle \lim_{x \to -2} f(x)$
2. b) $\displaystyle \lim_{x \to 0} f(x)$
3. c) $\displaystyle \lim_{x \to 3} f(x)$

Calculate the following direct limits by substituting directly into the expression for the function:

1. a) $\displaystyle \lim_{x \to 3} (x^2 - 5x + 2)$

1. b) $\displaystyle \lim_{x \to -1} \frac{2x + 4}{x^2 + 1}$

1. c) $\displaystyle \lim_{x \to 0} e^{x+1}$

In a company's cost analysis, it has been determined that the average cost of manufacturing $x$ units of a new tech gadget is given by the function:

Calculate the limit $\displaystyle \lim_{x \to +\infty} C(x)$ and interpret the result in the context of the problem (what happens to the cost per unit if production increases massively?).

Resolve the indeterminate form $\frac{\infty}{\infty}$ in the following cases by comparing the degrees of the polynomials:

1. a) $\displaystyle \lim_{x \to +\infty} \frac{4x^2 - 3x + 1}{2x^2 + 5}$

1. b) $\displaystyle \lim_{x \to +\infty} \frac{x + 10}{x^2 - 2}$

1. c) $\displaystyle \lim_{x \to +\infty} \frac{x^3 - 1}{100x^2 + x}$

Calculate the value of the following limit by resolving the indeterminate form $\frac{0}{0}$ through simplifying the fraction (taking out a common factor or factorising):

The number of followers of a beginner "streamer" channel during the first months follows the function $S(t) = \frac{200t^2}{t^2 + 4}$, where $t$ is the time in months. The graph shows the evolution of $S(t)$.

1. a) How many followers does the channel have at the start ($t=0$)?
2. b) Use limit calculation to find out how many followers the channel will tend to in the long term (when $t \to +\infty$).

Calculate the limit at infinity of the following rational function and justify your answer:

To calculate the limit $\lim_{x \to 2} \dfrac{x^2 - 4}{x^2 + 2x - 8}$, two classmates reason as follows:

• Hugo substitutes $x = 2$ and obtains $\dfrac{0}{0}$. He states: "Since zero divided by any number is zero, the limit is $0$".

• Mateo replies: "No, Hugo, any quantity divided by itself equals $1$, so the limit must be $1$".

1. a) Explain with sound reasoning the conceptual error that both are making and clarify why the expression $\left[\frac{0}{0}\right]$ is called an "indeterminate form" and does not represent a defined numerical value.
2. b) Apply polynomial factorisation using the remainder theorem (or Ruffini's rule), simplify the factor generating the indeterminate form, and find the true numerical value of the limit.

Given the function $f(x) = \dfrac{\sqrt{x + 7} - 3}{x - 2}$:

1. a) Estimate: Without performing algebraic transformations, evaluate numerically with your calculator the values of $f(1,99)$ and $f(2,01)$ to at least four decimal places. From the results obtained, propose a reasoned conjecture about what the exact value of $\lim_{x \to 2} f(x)$ should be, expressed as an irreducible fraction.
2. b) Check: Analytically calculate $\lim_{x \to 2} f(x)$ by multiplying the numerator and denominator by the conjugate expression of the radical to resolve the indeterminate form $\left[\frac{0}{0}\right]$. Check whether the theoretical value obtained matches your initial estimate exactly.

Given the graph of the function $f(x)$ shown below, analyse its behaviour at the critical points and determine the following values:

1. a) $\lim_{x \to -2^-} f(x)$ and $\lim_{x \to -2^+} f(x)$
2. b) $\lim_{x \to 1} f(x)$
3. c) $\lim_{x \to 3^-} f(x)$ and $\lim_{x \to 3^+} f(x)$
4. d) Does the overall limit exist at $x = 3$? Justify your answer based on the one-sided limits.

Calculate the value of the following limit by resolving the indeterminate form $\frac{0}{0}$ through polynomial factorisation (you may use Ruffini's rule if necessary):

A courier company has determined that its daily costs in euros are given by the function $C(x) = \sqrt{9x^2 + 5x} - 3x$, where $x$ represents the number of deliveries made in hundreds.

Calculate the limit of the costs as the delivery volume increases indefinitely ($\lim_{x \to +\infty} C(x)$). Interpret the result obtained in the context of the problem.

Calculate the following limits by analysing the degree of the polynomials and the highest power terms (indeterminate form $\frac{\infty}{\infty}$):

1. a) $\lim_{x \to +\infty} \frac{2x^2 - 5x + 1}{\sqrt{x^4 + 3x^2}}$

1. b) $\lim_{x \to -\infty} \frac{x^3 + 2}{x^2 + 10}$

Determine the value of the parameter $k \in \mathbb{R}$ so that the limit of the function $f(x)$ exists at the point $x = 1$, given the function:

For that value of $k$, what is the value of the limit?

Evaluate the following limit that leads to an indeterminate form of type $1^\infty$ using the formula related to the number $e$:

The number of followers of a new "influencer" on a social network follows the model given by the function $S(t) = 500 \cdot \left( 1 + \frac{2}{t+1} \right)^t$, where $t$ is the time in months since he opened his account.

1. a) Calculate $\lim_{t\to+\infty} S(t)$ and, from it, state the trend in the number of followers as time grows indefinitely. Justify your answer by calculating limits.

1. b) Write the equation of the horizontal asymptote of $S(t)$ as $t\to+\infty$ and interpret its meaning in the context of the problem.

Calculate the limit of the following rational function at the point $x = 1$. Note that if you obtain the form $\frac{k}{0}$, you must examine the one-sided limits to decide whether the limit is $+\infty$, $-\infty$ or does not exist:

To calculate the limit at infinity $\lim_{x \to +\infty} \left( \sqrt{4x^2 + 3x - 1} - 2x \right)$, two different strategies are proposed in the classroom:

• Method A (dominant term shortcut): "Inside the radical, as $x \to +\infty$, the dominant term is $4x^2$. Therefore, $\sqrt{4x^2 + 3x - 1} \approx \sqrt{4x^2} = 2x$. Subtracting $2x - 2x$, the limit is directly $0$".

• Method B (multiplication by the conjugate): "Since it is an indeterminate form $\infty - \infty$, the direct approximation hides first-order terms; you must multiply and divide the entire expression by its conjugate $(\sqrt{4x^2 + 3x - 1} + 2x)$ and then compare degrees".

1. a) Justify why Method A is not mathematically valid and leads to an incorrect result in this specific case.
2. b) Fully develop Method B, justifying each algebraic step, and calculate the actual value of the limit.
3. c) Based on the result, determine with sound reasoning whether the graph of the function $f(x) = \sqrt{4x^2 + 3x - 1}$ has an oblique asymptote of the form $y = 2x + n$ as $x \to +\infty$, stating the value of the y-intercept $n$.

A bank in Spain offers a one-year deposit with an initial principal of $5.000$ € at a nominal annual interest rate of $4\,\%$ ($r = 0,04$). If interest is compounded periodically $n$ times a year, the accumulated final capital is given by the compound interest formula $C(n) = 5.000 \cdot \left(1 + \dfrac{0,04}{n}\right)^n$.

1. a) A client states: "If we divide the year into infinitely many compounding periods ($n \to +\infty$), the fraction $\frac{0,04}{n}$ tends to $0$, so the base becomes $1$. Since $1$ raised to any number is $1$, in the end we will get $5.000 \cdot 1 = 5.000$ € and we will lose all the accumulated interest". Refute this reasoning with mathematical rigour and identify which indeterminate form actually appears.
2. b) Calculate the limit $\lim_{n \to +\infty} C(n)$ applying the base-$e$ method for the indeterminate form $1^\infty$ ($e^L$, with $L = \lim_{n \to +\infty} n \cdot (g(n) - 1)$).
3. c) Calculate the final capital with monthly compounding ($n = 12$) and compare it with the limit under continuous compounding ($n \to +\infty$), rounding both to the nearest euro cent. In light of the difference, evaluate whether it is worth it for the saver to demand continuous compounding over monthly compounding.

Given the rational function $f(x) = \dfrac{x^3 + ax^2 + bx + 2}{x^2 - 1}$, determine the values of the real parameters $a$ and $b$ knowing that the finite limit of the function exists as $x$ tends to $1$ and that this limit is exactly $L = -2$. Once you have found the parameters, justify with reasoning whether the function has a vertical asymptote at $x = -1$.

A chain of gyms estimates that the cumulative number of members (in thousands) follows the function $B(t) = \sqrt{4t^2 + 5t + 1} - (at + b)$, where $t$ is the time in years.

• a) Determine the value of $a$ so that, in the long term ($t \to \infty$), the number of members neither grows indefinitely nor becomes negative (that is, so that the limit is a finite real value).

• b) For the value of $a$ found in the previous part, calculate the value of $b$ required for the limit of the number of members as time tends to infinity to be exactly $3$ thousand.

Calculate the value of the following limit involving an indeterminate form of type $1^\infty$ in the context of continuous compound interest:

Then, explain what would happen to the value of the limit if the degree of the exponent were $2$ instead of $1$.

Study the behavior at infinity of the following function by calculating the limits as $x \to +\infty$ and as $x \to -\infty$:

Pay special attention to the sign change when you bring factors into or take them out of the root when the variable tends to minus infinity. Justify whether horizontal asymptotes exist in both directions and, if so, write their equations.

Given the piecewise function:

Calculate the value that the parameter $k$ should take so that the limit of the function exists at the point $x = 0$. Is it possible to assign a value to $g(0)$ so that the function is continuous at that point? Justify your answer using the concept of one-sided limits and indeterminate forms.

In the analysis of the spread of news on a social network, the function $P(t) = \dfrac{100}{1 + 20 \cdot e^{-0,5t}}$ is used, where $P$ is the percentage of the population aware of the news and $t$ is the time in hours.

• a) Calculate $\lim_{t \to \infty} P(t)$ and interpret the result in the context of the problem.

• b) If the rate of spread is given by the derivative (which we will study later, but suppose you are given the simplified expression for relative growth), analyse what happens to the difference $100 - P(t)$ as $t \to \infty$.

• c) Determine, with reasoning, whether the function has any vertical asymptotes.

Calculate the limit of the following expression by resolving the indeterminate form $\infty - \infty$ using the conjugate:

If the result of the limit is $L = 4$, what must the value of the parameter $a$ be? Compare this result with the limit of the same function as $x \to -\infty$.

Analyse the hierarchy of infinities to evaluate the following limit, justifying each step according to the rate of growth of the functions involved (exponential, polynomial and logarithmic):

Would the result change if the base of the exponential in the denominator were $e$ instead of $3$? Justify your answer based on the comparison of orders of magnitude.

The analytics department of an e-commerce platform models the purchasing behaviour of its users using the rational function $f(x) = \dfrac{a x^2 + b x + c}{x^2 - 5x + 6}$, where $x$ represents the time spent on the website (in minutes) and $a, b, c \in \mathbb{R}$ are constant parameters.

1. a) Deduce with sound reasoning the value of the coefficients $a$, $b$, and $c$, knowing that the following two conditions are met simultaneously:

1) When the browsing time extends indefinitely ($x \to +\infty$), the value of the function stabilises at a finite value equal to $3$.

2) At the critical moment $x = 2$, the function generates an indeterminate form of the type $\left[\frac{0}{0}\right]$, but the platform does not crash because the finite limit $\lim_{x \to 2} f(x) = -1$ exists.

1. b) Once the parameters are determined, study what happens at the other point where the denominator vanishes ($x = 3$). Calculate the one-sided limits $\lim_{x \to 3^-} f(x)$ and $\lim_{x \to 3^+} f(x)$, and interpret the situation from the perspective of the function's vertical asymptotes.

The National Statistics Institute (INE) prepares long-term demographic projections to assess the risk of depopulation in two rural regions in inland Spain. The mathematical models project the population $P(t)$ (in thousands of inhabitants) as a function of elapsed time $t \ge 0$ (in decades elapsed since the year 2024):

• Region A: $P_A(t) = \sqrt{9t^2 + 12t + 5} - 3t$

• Region B: $P_B(t) = 5 \cdot \left(\dfrac{t + 1}{t + 2}\right)^t$

1. a) Analyse the very long-term trend ($t \to +\infty$) for Region A. Identify the initial indeterminate form, resolve it justifying the algebraic procedure, and calculate what number of inhabitants it tends to stabilise at.
2. b) Analyse the very long-term trend ($t \to +\infty$) for Region B. Identify the type of indeterminate form, resolve it in detail using the power rule related to the number $e$, and find the limiting population in inhabitants.
3. c) Considering that in the year 2024 ($t = 0$) Region A had about $2.236$ inhabitants ($P_A(0) = \sqrt{5}$) and Region B had $5.000$ inhabitants ($P_B(0) = 5$), calculate what percentage of its original population each region retains in the long run and justify which of the two territories exhibits more resilient demographic behaviour.