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, specify this 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 when 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 of type $\left[\frac{\infty}{\infty}\right]$ in the following limits of rational functions, comparing the degrees of the numerator and the 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 $f(x) = \frac{x^2 - 9}{x - 3}$ when $x$ tends to $3$.

Hint: Note that by substituting directly we obtain an indeterminate form of type $\left[\frac{0}{0}\right]$. Try to factorize the numerator to simplify the expression.

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

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

Indicate with reasoning 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 bacteria population 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.

What will the bacteria population be if the experiment is prolonged indefinitely? To do this, calculate $\lim_{t \to +\infty} P(t)$.

Calculate the value of the following limit which results in the indeterminate form of type $[1^\infty]$ using the abbreviated formula for the number $e$:

Calculate the following limit by resolving the indeterminate form of type $\frac{0}{0}$ 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. Pay special attention to the powers inside the roots:

Resolve the following indeterminate form of type $\infty - \infty$. To do this, perform the prior 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. Calculate the value the number of users tends to when the time increases indefinitely ($t \to +\infty$).

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

Given the following graph of the function $f(x)$, visually analyze and analytically calculate the one-sided limits at the points of discontinuity to determine if 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 present a jump.

Use the comparison of orders of infinity to solve the following limits without the need for extensive operations. 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 results in an indeterminate form of type $1^\infty$. Express the result in terms of the number $e$:

Calculate the value of the following limit by analysing the hierarchy of the infinities involved. You must justify which term dominates in the numerator and in the denominator before giving the final result:

Determine the values of the real parameters $a$ and $b$ so that the following limit is a finite real number not equal to zero. Once found, calculate the value of this limit:

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

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

Consider the piecewise function:

Find the values of $a$ and $k$ so that the one-sided limits exist at $x=1$ 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 type $\infty - \infty$ with roots of different orders:

Hint: Add and subtract a suitable intermediate term to separate the limit into two manageable parts.

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 behavior of the pressure when the speed $v$ approaches $3$ ft/s. If the limit is indeterminate, resolve it and indicate whether the pressure tends to a stable value.

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