Exercises: Indeterminate forms

Observe the following graph of the function $f(x)$ and determine, intuitively, the value of the indicated limits. If any limit does not exist or is infinite, state so clearly.

1. a) $\lim_{x \to -2} f(x)$
2. b) $\lim_{x \to 0} f(x)$
3. c) $\lim_{x \to 3} f(x)$
4. d) $\lim_{x \to +\infty} f(x)$

Calculate the following limits of polynomial functions as the variable $x$ tends to infinity. Remember to identify the dominant term in each case:

1. a) $\lim_{x \to +\infty} (-2x^3 + 5x^2 - 1)$
2. b) $\lim_{x \to -\infty} (x^4 - 100x^2)$
3. c) $\lim_{x \to +\infty} \frac{3x^2 - 5}{10}$

Resolve the indeterminate form $\left[\frac{\infty}{\infty}\right]$ in the following limits of rational functions by comparing the degrees of the numerator and denominator:

1. a) $\lim_{x \to +\infty} \frac{5x^2 - 3x + 2}{2x^2 + 1}$
2. b) $\lim_{x \to +\infty} \frac{x + 5}{x^2 - 3}$
3. c) $\lim_{x \to +\infty} \frac{x^3 - 1}{x^2 + 10x}$

Calculate the limit of the function $f(x) = \frac{x^2 - 9}{x - 3}$ as $x$ tends to $3$.

Help: Notice that direct substitution gives an indeterminate form of the type $\left[\frac{0}{0}\right]$. Try factorising the numerator to simplify the expression.

Given the rational function $f(x) = \frac{2}{x-4}$, study its behaviour near its vertical asymptote by calculating the one-sided limits:

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

State with reasons whether the overall limit exists at $x = 4$.

In the study of computer algorithms, it is crucial to know which function grows faster. Based on the hierarchy of infinities, assign the value of the following limits ($0$, $+\infty$ or $-\infty$):

1. a) $\lim_{x \to +\infty} \frac{\ln(x)}{x}$
2. b) $\lim_{x \to +\infty} \frac{e^x}{x^{10}}$
3. c) $\lim_{x \to +\infty} \frac{x^2 + 1}{\ln(x)}$

A team of scientists is monitoring the growth of a population of bacteria in a laboratory. The population (in millions) follows the model given by the function:

where $t$ is the time in hours since the start of the experiment. The graph shows the evolution of the population during the first 20 hours.

What will the bacteria population be if the experiment were extended indefinitely? To find out, calculate $\lim_{t \to +\infty} P(t)$.

Calculate the value of the following limit leading to an indeterminate form of the type $[1^\infty]$ using the shortcut formula for the number $e$:

Mateo is solving the limit $\lim_{x \to +\infty} \left(\frac{x+5}{x+2}\right)^{3x}$ on an exam and writes the following in his solution:

"Since $\lim_{x \to +\infty} \frac{x+5}{x+2} = 1$ and the number $1$ raised to any power is always $1$, the limit presents no difficulty and is directly equal to $1^\infty = 1$".

1. a) Explain in your own words why Mateo's reasoning is conceptually flawed. In your answer, you must clarify what it strictly means for an expression to be an "indeterminate form" and which two opposing tendencies conflict in this specific case.
2. b) Correct the exercise by showing the formal algebraic procedure and calculate the true, exact value of the limit.

Given the function $f(x) = \frac{4 \cdot 3^x - 5x^6}{2 \cdot 3^x + 7x^2 + 1}$:

1. a) Before performing any formal algebraic calculation, estimate the value of $\lim_{x \to +\infty} f(x)$ relying exclusively on the hierarchy of infinites. Justify which terms are dominant in the numerator and in the denominator when $x$ takes sufficiently large values, and what the relative weight of powers is compared to exponential functions.
2. b) Formally verify your estimate by calculating the limit by factoring out the dominant term in both the numerator and the denominator to resolve the indeterminate form $\frac{\infty}{\infty}$.
3. c) Reason whether the result obtained in the previous section would remain the same if the limit were evaluated towards $-\infty$, that is, when calculating $\lim_{x \to -\infty} f(x)$. Justify your conclusion by analyzing the behavior of $3^x$ as $x \to -\infty$.

Calculate the following limit by resolving the indeterminate form $\frac{0}{0}$ using the conjugate technique:

Determine the value of the limit at infinity by analysing the degree of the dominant terms in the numerator and the denominator. Take special care with the powers inside the roots:

Resolve the following indeterminate form $\infty - \infty$. To do so, perform the preliminary algebraic operation and simplify the resulting expression:

In the development of a new social network, the number of active users (in millions) follows the function $U(t) = \left(\frac{2t+3}{2t-1}\right)^{t+1}$, where $t$ is the time in months since its launch (with $t>\frac{1}{2}$). Calculate the value the number of users approaches as time grows indefinitely ($t \to +\infty$).

Find the values of the real parameters $a$ and $b$ so that the following limit is finite and equals $4$:

Given the following graph of the function $f(x)$, analyse visually and calculate analytically the one-sided limits at the points of discontinuity to determine whether the overall limit exists at $x = 2$:

Calculate the value of $a$ so that the limit at $x = 2$ is unique and the function does not have a jump.

Use comparison of orders of infinity to solve the following limits without performing lengthy calculations. Justify your answer by indicating which function "dominates" the other:

1. a) $\lim_{x \to +\infty} \frac{e^x + x^5}{100x^6 - \ln(x)}$

1. b) $\lim_{x \to +\infty} \frac{\sqrt{x^7 + 1}}{x^3 \cdot \ln(x)}$

Calculate the value of the following exponential limit which leads to an indeterminate form of the type $1^\infty$. Express the result in terms of the number $e$:

A sociology team uses projections from the National Statistics Institute (INE) to model the demographic evolution of a rural region in Aragón. The population $P(t)$, expressed in thousands of inhabitants, is estimated as a function of time $t$ (in years elapsed since 2024) using the function:

1. a) Analyze the long-term demographic trend ($t \to +\infty$) by identifying what type of indeterminate form arises in each of the two terms of $P(t)$.
2. b) Reasonably resolve both indeterminate forms using the appropriate techniques (degree comparison in the rational function and the conjugate technique in the radical expression).
3. c) Interpret the numerical value obtained for $\lim_{t \to +\infty} P(t)$: does the region tend toward complete depopulation, indefinite growth, or demographic stability? State what specific number of inhabitants the region will tend toward according to this model.

To solve the limit $\lim_{x \to +\infty} \left( \frac{x^2 + 4x}{x^2 + x + 1} \right)^{2x}$, two 2nd year of Bachillerato students propose different approaches after noticing that it is an indeterminate form of type $1^\infty$:

• Method A (Canonical construction of Euler's number $e$): Express the base in the form $1 + \frac{1}{g(x)}$, force the presence of $g(x)$ in the exponent by multiplying and dividing to apply the fundamental definition $\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^u = e$, and compute the limit of the resulting exponent.

• Method B (Direct power formula): Use the general identity $\lim_{x \to a} [u(x)]^{v(x)} = e^{\lim_{x \to a} v(x) \cdot (u(x) - 1)}$.

1. a) Solve the limit by developing all the steps of Method A in detail.
2. b) Solve the same limit applying Method B and confirm that both results match.
3. c) Compare both approaches by analyzing the algebraic calculation workload and the risk of making sign errors when manipulating algebraic fractions. Justify which of the two methods is more efficient and safer in a time-limited test such as the EBAU/PAU.

Calculate the value of the following limit by analysing the hierarchy of the infinities involved. Give reasons to justify which term dominates in the numerator and in the denominator before providing the final result:

Determine the values of the real parameters $a$ and $b$ so that the following equality holds:

That is, the limit must be a finite real number and equal $-\dfrac{1}{2}$. Once you have found $a$ and $b$, check that the limit really equals $-\dfrac{1}{2}$.

In the design of a new telecommunications antenna, the signal intensity $I$ as a function of the distance $d$ to the repeater is given by the expression:

Calculate the limiting signal intensity as distance $d$ tends to infinity (indeterminate form of the type $1^\infty$).

Solve the following limit using the double conjugate technique to resolve the $\frac{0}{0}$ indeterminate form:

Let the function be defined piecewise:

Find the values of $a$ and $k$ so that the one-sided limits at $x=1$ exist and are equal to $2$. What type of discontinuity does the function have at $x=1$ if we do not define $f(1)$?

Calculate the value of the following expression which presents an indeterminate form of the type $\infty - \infty$ with roots of different orders:

A team of aerospace engineers is studying the pressure $P$ (in pascals) inside a ventilation duct. The pressure depends on the airflow speed $v$ according to the formula:

Study the behaviour of the pressure as the speed $v$ approaches $3$ m/s. If the limit is indeterminate, resolve it and state whether the pressure tends to a stable value.

Given the function $g(x) = \frac{|x-1|}{x^2-1} \cdot e^{\frac{1}{x-1}}$, analyse the existence of the one-sided limits as $x \to 1$. Identify the indeterminate forms that appear on each side and classify the type of discontinuity at that point.

Consider the function dependent on two real parameters $a$ and $b$ defined by $f(x) = \sqrt{9x^2 - 12x + 7} - (ax + b)$.

1. a) Determine the values of the parameters $a$ and $b$ so that the following equality holds:

1. b) Analytically prove that if $a \ne 3$, it is impossible for the limit of $f(x)$ as $x \to +\infty$ to be a finite value. Reasonably distinguish the cases $a < 3$ (differentiating whether $a \le 0$ or $0 < a < 3$) and $a > 3$, explaining the result obtained in each situation.
2. c) Setting the value of $a$ found in section a), calculate what value the parameter $b$ must take for the limit to be exactly $-4$, that is, $\lim_{x \to +\infty} f(x) = -4$.

Consider the following limit involving roots of different indices:

1. a) [Estimate]: Perform an intuitive asymptotic approximation before manipulating algebraically. Knowing that for very large values of $x$ it holds that $\sqrt[3]{8x^3+12x^2} = 2x\left(1+\frac{3}{2x}\right)^{1/3}$ and $\sqrt{4x^2-8x} = 2x\left(1-\frac{2}{x}\right)^{1/2}$, and using the linear approximation $(1+u)^\alpha \approx 1 + \alpha u$ as $u \to 0$, reasonably conjecture the final value of the limit.
2. b) [Verify]: Formally solve the limit using the intermediate term technique: introduce $C(x) = 2x$ by adding and subtracting this expression to split the limit into the sum of two independent indeterminate forms of type $\infty - \infty$:

Solve each block by applying the conjugate corresponding to its index (the identity $A^3 - B^3 = (A-B)(A^2 + AB + B^2)$ for the cube root and the standard quadratic conjugate for the square root).

1. c) Compare the analytical result with the conjecture obtained in section a) and explain why choosing any other linear intermediate term different from $2x$ (for example, taking $C(x) = x$ or $C(x) = 3x$) would not allow resolving the overall indeterminate form.