Exercises: Calculation of limits

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

1. a) $\lim_{x \to 2^-} f(x)$

1. b) $\lim_{x \to 2^+} f(x)$

1. c) $\lim_{x \to +\infty} f(x)$

Calculate the limit at infinity of the following polynomial function, which represents the evolution of the profits of a tech startup (in thousands of dollars) over time $x$:

$$\lim_{x \to +\infty} ( -2x^3 + 50x^2 - 100 )$$

Indicate which is the dominant term and justify why the sign of the leading coefficient determines the final behavior of the function.

A video game server manages the number of connected players using the function $P(x) = \frac{6x^2 + 20}{2x^2 + 5}$, where $x$ is the time in hours since the launch of an event. Calculate the limit of players as time tends to infinity:

$$\lim_{x \to +\infty} \frac{6x^2 + 20}{2x^2 + 5}$$

In a chemistry laboratory, the concentration of a reactant in a mixture (in mg/l) follows the function $C(t) = \frac{5t + 8}{t^2 + 2}$, where $t$ is the time in minutes. Determine what value the concentration tends to in the long term by calculating:

$$\lim_{t \to +\infty} \frac{5t + 8}{t^2 + 2}$$

Calculate the one-sided limits of the function $f(x) = \frac{3}{x - 5}$ at the point where the denominator equals zero. To do this, complete the study of the trend from the left and from the right:

1. a) $\lim_{x \to 5^-} \frac{3}{x - 5}$

1. b) $\lim_{x \to 5^+} \frac{3}{x - 5}$

Use the representation of the vertical asymptote to help you:

Resolve the indeterminate form of type $\frac{0}{0}$ by simplifying the fraction through the factorization of the polynomials (you can use algebraic identities or Ruffini's rule):

$$\lim_{x \to 2} \frac{x^2 - 4}{x^2 - 2x}$$

Based on the comparison of orders of infinity, determine the value of the following limits without performing complex operations:

1. a) $\lim_{x \to +\infty} \frac{e^x}{x^{10}}$

1. b) $\lim_{x \to +\infty} \frac{\ln(x)}{x^2}$

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

The altitude of a reconnaissance drone (in feet) with respect to the horizontal distance $x$ traveled is given by the function $h(x) = \frac{x^2 - 9}{x - 3}$. Although the drone cannot physically be at position $x = 3$ due to a software restriction, calculate the limit of the altitude as it approaches this point:

$$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$

A precision sensor in a nanotechnology laboratory measures the force of attraction between two particles. The signal strength $S$ as a function of the distance $x$ (in nanometers) is expressed by the following function:

$$S(x) = \frac{\sqrt{x+5} - 3}{x-4}$$

Calculate the value to which the signal tends as the distance $x$ approaches $4$ nanometer. Explain what mathematical technique you have used to resolve the indeterminate form.

In the study of the spread of news on social media, it is estimated that the number of users reached $U$ (in thousands) after $t$ hours follows the expression:

$$U(t) = \frac{5t^2 - 3t + 2}{2t^2 + \sqrt{t^4 + 1}}$$

Calculate the limit of users reached as time increases indefinitely ($t \to +\infty$). Does the function have any horizontal asymptotes? Justify your answer by comparing the degrees of the dominant terms.

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

$$\lim_{x \to +\infty} \left( \sqrt{x^2 + 6x} - x \right)$$

Graphically represent in a schematic way what this result means regarding the distance between the radical function and the line $y = x$.

Given the rational function depending on a parameter $k \in \mathbb{R}$:

$$f(x) = \frac{x^3 - kx^2 + 4}{x-2}$$

Find the value of $k$ so that the limit $\lim_{x \to 2} f(x)$ is finite. Once this value is found, calculate the value of the limit.

The growth of a bird population in a protected nature reserve is governed by the following idealised continuous compound interest law:

$$P(t) = \left( \frac{3t + 5}{3t - 2} \right)^{2t}$$

Calculate the limit of the population as time $t$ tends to infinity. Express the result in terms of the number $e$.

Calculate the value of the following limits by comparing the orders of infinity of the functions involved (exponentials, powers and logarithms):

1. a) $\lim_{x \to +\infty} \frac{\ln(x^{10} + 5)}{x}$

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

1. c) $\lim_{x \to +\infty} \frac{e^x + x^5}{2e^x - \sqrt{x}}$

Consider the piecewise function $f(x)$:

$$f(x) = \begin{cases} \frac{x^2 - 9}{x - 3} & \text{if } x < 3 \\ \sqrt{2x + a} & \text{if } x \ge 3 \end{cases}$$

Calculate the value of the parameter $a$ so that the limit of the function exists at the point $x = 3$. What is the value of this limit?

Calculate the limit at the point $x = 0$ of the following trigonometric expression, using equivalent infinitesimals or known trigonometric identities:

$$\lim_{x \to 0} \frac{x \cdot \text{tg}(3x)}{\cos(x) - 1}$$

Hint: Remember that for very small values of $x$, $\text{tg}(ax) \approx ax$ and $1 - \cos(x) \approx \frac{x^2}{2}$.

A multiplayer video game server has detected that the number of active users, $N(t)$, in thousands, follows an exponential growth model as a function of time $t$ (in months) defined by the expression:

$$N(t) = \left( \frac{3t^2 + 5t + 1}{3t^2 + 2t} \right)^{2t}$$

Calculate the limit of active users as time tends to infinity ($t \to +\infty$). Does the player population stabilise at a finite value or grow indefinitely?

In the design of a new ramp for a skatepark, engineers need to ensure that the transition between two sections of the track is smooth (continuous). The height of the ramp $h(x)$ in feet is given by the following function:

$$h(x) = \begin{cases} \dfrac{\ln(1 + kx)}{x} & \text{if } x > 0 \\ \cos(x) + a & \text{if } x \le 0 \end{cases}$$

Determine the values of the parameters $k$ and $a$ so that $\displaystyle \lim_{x \to 0} h(x)$ exists and is equal to the initial height of the ramp, which is $2$ feet.

Calculate the value of the following limit by applying the comparison of infinities or L'Hôpital's Rule, analysing in detail the order of magnitude of each term:

$$\lim_{x \to +\infty} \frac{e^x + x^{10}}{2e^x - \ln(x^5 + 1)}$$

Which term dominates the growth of the function in the numerator and which in the denominator?

A satellite follows a trajectory whose radial distance to the center of a planet, in thousands of miles, is given by $d(x) = \sqrt{x^2 + ax} - \sqrt{x^2 - 1}$, where $x$ represents the mission time. Knowing that after a long time ($x \to +\infty$) the distance stabilizes at $4$ thousand miles, determine the value of the orbital constant $a$.

Determine with reasoning the value of the parameters $a$ and $b$ (where $a, b \in \mathbb{R}$) so that the following equality of limits holds:

$$\lim_{x \to 0} \frac{ax^2 + bx + 1 - e^x}{x^2} = 3$$

Hint: Note that for the limit to be finite, the numerator must equal zero at the point in the same way as the denominator.

Study the behavior of the function $f(x) = \left( \cos x \right)^{1/x^2}$ as $x$ approaches zero. To do this, calculate:

$$\lim_{x \to 0} (\cos x)^{1/x^2}$$

Explain why this limit results in an indeterminate form of the type $1^\infty$ and solve using natural logarithms or the specific formula for this type of limit.

Let the function be $g(x) = \dfrac{x^3 - 2x^2 + k}{x^2 - 4}$. Calculate the value of $k$ so that the limit of the function as $x \to 2$ is finite. Once $k$ is found, calculate this limit and classify the type of discontinuity the function presents at $x = 2$ and at $x = -2$.

Calculate the value of the limit of the following sequence, which frequently appears in continuous compound interest and probability problems:

$$\lim_{n \to \infty} \left( \sqrt{n^2 + n} - n \right)^n$$

Note: This exercise requires careful treatment of the internal indeterminate form before evaluating the overall power.