Exercises: Calculation of limits of functions

The temperature of an electronic component (in degrees Fahrenheit) 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.

The number of bacteria in a culture follows the growth model $B(x) = -2x^3 + 50x^2 + 1000$, where $x$ is the number of days elapsed. Determine the limit of $B(x)$ as $x \to +\infty$ and state the long-term behavior of this culture according to the mathematical model.

The average production cost of $x$ thousand units of an item is given by the function $C(x) = \frac{4x^2 + 150x}{2x^2 + 5}$. Calculate the limit of the average cost as the number of units produced tends to infinity.

Calculate the following algebraic limit by resolving the indeterminate form through factorization:

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

Analyze the one-sided limits of the function $f(x) = \frac{2x + 1}{x - 4}$ as $x$ tends to $4$ and determine whether the overall limit exists at that point.

Evaluate the limit of the following rational function as the independent variable tends to minus infinity:

$\lim_{x \to -\infty} \frac{7x^4 - 3x^2 + 1}{5x^3 + 2x - 8}$

The speed of a test vehicle (in ft/s) as a function of time $t$ (in seconds) is defined piecewise as $v(t) = 3t + 2$ if $0 \le t < 4$, and $v(t) = t^2 - 2$ if $t \ge 4$. Determine using one-sided limits whether the speed function is continuous at time $t = 4$.

Calculate the limit of the exponential function $h(x) = 5 \cdot \left(\frac{2}{3}\right)^x + 8$ as $x$ tends to $+\infty$.

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

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

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

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

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

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

An electricity company sets its tariff $T(x)$ in dollars as a function of consumption $x$ in kWh. For consumptions less than $100$ kWh, the tariff is $T(x) = 0,15x + 5$. For consumptions 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 the consumption approaches $100$ kWh.

Calculate the limit at infinity of the following function containing radicals, by analyzing the degrees of the numerator and the denominator:

$\lim_{x \to +\infty} \frac{\sqrt{9x^2 + 4x} - 2x}{5x + 1}$

The concentration of a medicine in the bloodstream (in mg/L) after $t$ hours of its administration is given by $C(t) = \frac{25t}{t^2 + 4}$. If the patient does not take any additional doses, find the limit of the concentration as time tends to infinity and interpret the result.

Calculate the value of the parameter $k$ so that the following equality is satisfied in the limit:

$\lim_{x \to +\infty} \frac{kx^3 - 5x + 1}{2x^3 + x^2 - 8} = 4$

Analyze the existence of the limit at $x = 0$ of the function $f(x) = \frac{|x|}{x} + x^2$ by first calculating its one-sided limits.

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

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

Determine the values of the real parameters $a$ and $b$ so that the following limit condition is satisfied:

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

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

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

Consider a rectangle inscribed in a circle of radius $R = 10$ inches. If $x$ is the length of the base of the rectangle, its area is given by the function $A(x) = x \sqrt{400 - x^2}$. An architect needs to know how the ratio between the area of the rectangle and the square of its base behaves when the base becomes very small. Calculate $\lim_{x \to 0^+} \frac{A(x)}{x^2}$.

Calculate the parametric limit $\lim_{x \to +\infty} \left( \frac{2x^2 + 3}{2x^2 - x} \right)^{ax+1}$ and determine the value of $a$ so that the exact result is $e^2$.

The relativistic mass of a moving particle is given by $m(v) = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $m_0$ is the rest mass, $v$ is the speed of the particle and $c$ is the speed of light. Calculate the limit of the mass as the speed $v$ tends to the speed of light $c$ from the left ($v \to c^-$) and interpret the mathematical result in the physical context.

Calculate the following limit at infinity involving exponential functions with different bases:

$\lim_{x \to +\infty} \frac{3^{x+1} - 5^x}{2^x + 5^{x-1}}$

Let the rational function be $f(x) = \frac{x^3 - 8}{x^2 - 4}$. Find the equation of the oblique asymptote line $y = mx + n$ of this function by rigorously calculating the limits $m = \lim_{x \to \infty} \frac{f(x)}{x}$ and $n = \lim_{x \to \infty} (f(x) - mx)$.