Calculating limits

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

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

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.

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

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

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

Study the behaviour 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.

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?

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 $x = -2$.

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.

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