Calculating limits of functions

The temperature of an electronic component (in degrees Celsius) as a function of time $t$ (in hours) is given by the expression $T(t) = \frac{3t^2 - 5t + 12}{t + 2}$. Calculate the limit of the temperature as $t$ approaches $3$ hours.

Calculate the following algebraic limit by resolving the indeterminate form using factorisation:

$\lim_{x \to 5} \frac{x^2 - 25}{x^2 - 7x + 10}$

Resolve the following indeterminate form by factorising the polynomials:

$\lim_{x \to 3} \frac{x^2 - 9}{x^2 - 5x + 6}$

Calculate the value of the limit presenting an indeterminate form of the subtraction of algebraic fractions type:

$\lim_{x \to 2} \left( \frac{1}{x - 2} - \frac{4}{x^2 - 4} \right)$

Evaluate the following limit by multiplying the numerator and denominator by the corresponding conjugate expression:

$\lim_{x \to 3} \frac{\sqrt{x + 6} - 3}{x - 3}$

Evaluate the following limit with indeterminate form $\frac{0}{0}$ containing two radicals in the numerator:

$\lim_{x \to 0} \frac{\sqrt{4 + x} - \sqrt{4 - x}}{3x}$

Calculate the following limit, which presents an indeterminate form of type $\infty - \infty$ involving square roots:

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

Given the following piecewise function, calculate the left- and right-hand limits at $x = 2$ and determine whether $\lim_{x \to 2} f(x)$ exists:

$f(x) = \begin{cases} 2x + 1 & \text{if } x < 2 \\ x^2 + 1 & \text{if } x \ge 2 \end{cases}$

An electricity company sets its tariff $T(x)$ in euros based on electricity consumption $x$ in kWh. For consumption less than $100$ kWh, the tariff is $T(x) = 0,15x + 5$. For consumption greater than or equal to $100$ kWh, the tariff changes to $T(x) = \frac{ax^2 + 200}{x}$. Find the value of the parameter $a$ so that the limit of the tariff exists as consumption approaches $100$ kWh.

Determine the value of the following limit by resolving the indeterminate form associated with the number $e$:

$\lim_{x \to +\infty} \left( \frac{x + 5}{x - 2} \right)^{3x}$