Calculating 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 one-sided limits of the function $f(x) = \frac{3}{x - 5}$ at the point where the denominator vanishes. To do so, study the behavior of $f(x)$ as you approach from the left and from the right, paying attention to the sign of the numerator and the denominator:

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

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

The diagram shows the axes, the vertical asymptote $x=5$ and the direction in which we approach it from each side:

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

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

The altitude of a reconnaissance drone (in meters) with respect to the horizontal distance $x$ covered 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 that point:

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

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

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

The figure shows the graph of $S(x)$: for $x=4$ the function is not defined (point with the empty circle).

Calculate the value the signal approaches as the distance $x$ gets closer to $4$ nanometers. Explain which mathematical technique you used to resolve the indeterminate form.

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

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

Sketch a graph showing what this result means regarding the distance between the radical function and the line $y = x$.

You can do it on paper and send a photo with your answer.

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

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

$$\lim_{x \to +\infty} ( -120x^3 + 3 000x^2 - 6 000 )$$

State which is the dominant term and justify why the sign of the leading coefficient determines the end 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 the long-term value to which the concentration tends by calculating:

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

In the study of news propagation on social networks, 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 a horizontal asymptote? Justify your answer by comparing the degrees of the dominant terms.

Calculate 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 inner indeterminacy before evaluating the overall power.

Based on comparing 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) $\lim_{x \to +\infty} \frac{2^x}{3^x}$

Calculate the value of the following limits by comparing the orders of magnitude at 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) $\lim_{x \to +\infty} \frac{e^x + x^5}{2e^x - \sqrt{x}}$

Calculate the value of the following limit using 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?

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

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

Give reasons to determine the value of the parameters $a$ and $b$ (where $a, b \in \mathbb{R}$) such that the following equality of limits holds:

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

The growth of a bird population in a protected nature reserve is governed by the following idealised continuous compound growth 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 $e$.

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?

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

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

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

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.

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 $x = 3$. What is the value of this limit?

When designing 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 meters 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 $a$ such that $\displaystyle \lim_{x \to 0} h(x)$ exists and is equal to the initial height of the ramp, which is $2$ meters.

Let the function be $g(x) = \dfrac{x^3 - 2x^2 + k}{x^2 - 4}$.

1. a) Calculate the value of $k$ so that the limit of the function as $x \to 2$ is finite.

1. b) Once you have found $k$, calculate that limit.

1. c) Classify the type of discontinuity that the function has at $x = 2$ and at $x = -2$.