Exercises: Calculation of limits. Indeterminate forms

To understand the concept of a limit, let us analyse the behavior 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:

Given the graph of the function $f(x)$ shown below, analyse its behavior 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 pesos 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 $S(t) = 500 \times \left( 1 + \frac{2}{t+1} \right)^t$, 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:

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 $P(t) = \dfrac{100}{1 + 20 \times e^{-0.5t}}$ 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.