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 behavior 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 $e$:

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

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 $e$:

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 behavior 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 $g(x) = \frac{|x-1|}{x^2-1} \times e^{\frac{1}{x-1}}$, 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.