Limit of a Function as x → ∞

Calculate the value of the following polynomial limits as $x \to +\infty$ and as $x \to -\infty$:

1. a) $\lim_{x \to +\infty} (-2x^3 + 5x^2 - x + 7)$
2. b) $\lim_{x \to -\infty} (4x^4 - 3x^3 + 2)$

Determine the value of the following limits of rational functions where the degree of the numerator and the denominator are equal:

1. a) $\lim_{x \to +\infty} \frac{6x^2 - 3x + 1}{2x^2 + 5}$
2. b) $\lim_{x \to -\infty} \frac{-5x^3 + x}{4x^3 - 2x^2 + 1}$

Find the limit of the following rational functions given that the degree of the denominator is greater than that of the numerator:

1. a) $\lim_{x \to +\infty} \frac{3x + 8}{x^2 - 4x + 2}$
2. b) $\lim_{x \to -\infty} \frac{x^2 - 1}{x^3 + x^2 - 5}$

Find the value of the following limits where the degree of the numerator is greater than that of the denominator:

1. a) $\lim_{x \to +\infty} \frac{x^3 - 2x^2 + 1}{5x^2 - 3}$
2. b) $\lim_{x \to -\infty} \frac{-2x^4 + x}{x^2 + 3x - 1}$

The concentration of a drug in the bloodstream, in milligrams per liter, $t$ hours after administration is given by the function $C(t) = \frac{15t}{t^2 + 4}$. Calculate the limit of the concentration as $t \to +\infty$ and interpret the result in the context of the medical problem.

Find the equation of the horizontal asymptote of the function $f(x) = \frac{8x - 5}{2x + 3}$ by calculating the corresponding limits as $x \to +\infty$ and $x \to -\infty$.

Look at the graph of the function $f(x)$ and visually determine the limits $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$. From these, write the equations of the horizontal asymptotes of the function.

Calculate the limits of the following exponential functions as $x \to +\infty$ and as $x \to -\infty$:

1. a) $\lim_{x \to +\infty} \left(\frac{3}{5}\right)^x$
2. b) $\lim_{x \to -\infty} 4^x$

Calculate the limit of the following function which combines exponential and polynomial terms. Take into account which terms grow faster as $x \to +\infty$:

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

Given the piecewise-defined function $f(x)$, which equals $\frac{3x^2 - 1}{x^2 + 2}$ if $x \le 0$, and equals $\frac{2e^x + 5}{e^x - 1}$ if $x > 0$. Calculate the horizontal asymptotes of the function by determining the limits $\lim_{x \to -\infty} f(x)$ and $\lim_{x \to +\infty} f(x)$.

Calculate the following limit involving a square root in the numerator:

$\lim_{x \to +\infty} \frac{\sqrt{9x^2 + 4x - 1}}{3x + 2}$

Calculate the following limit as $x \to -\infty$, paying special attention to the sign of the variable when taking it out of the square root:

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

Calculate the following limit at negative infinity, which requires rationalisation and careful sign analysis when dealing with the implicit absolute value in the square root:

$\lim_{x \to -\infty} x \left( \sqrt{x^2 + 4} + x \right)$

Solve the following limit which gives an indeterminate form of the type $\infty - \infty$ by first subtracting the algebraic fractions:

$\lim_{x \to +\infty} \left( \frac{x^2 + 2x}{x - 1} - \frac{x^2 - 3}{x + 2} \right)$

Calculate the following limit using the properties of logarithms to first resolve the indeterminate form $\infty - \infty$:

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

Calculate the value of the following limit involving radicals, resolving the indeterminate form $\infty - \infty$ by multiplying and dividing by the conjugate expression:

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

Determine the value of the parameter $a$ so that the following limit, which presents an indeterminate form $\infty - \infty$, gives a result of $4$:

$\lim_{x \to +\infty} (\sqrt{x^2 + ax + 5} - \sqrt{x^2 - 2x + 1}) = 4$

Find the value of the following limit which gives an indeterminate form of the type $1^\infty$, expressing the final result in terms of the number $e$:

$\lim_{x \to +\infty} \left( \frac{2x + 5}{2x - 3} \right)^{4x}$

Find the value of the parameter $b$ so that the following equality holds, corresponding to a limit of the type $1^\infty$:

$\lim_{x \to +\infty} \left( \frac{x^2 + bx}{x^2 + 3} \right)^{2x} = e^6$

Determine the value of the parameter $k$ so that the following equality holds:

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

Find the values of the parameters $m$ and $n$ so that the graph of the function $f(x) = \frac{2x^2 - x + 3}{x + 1}$ has the line $y = mx + n$ as an oblique asymptote. To do this, set up and evaluate the limits $\lim_{x \to +\infty} \frac{f(x)}{x} = m$ and $\lim_{x \to +\infty} (f(x) - mx) = n$.

Let $f(x) = \frac{3x}{x+2}$ be the rational function. Calculate its limit $L$ as $x \to +\infty$. Then, set up and solve an inequality to find out from which value of $x > 0$ the distance between $f(x)$ and its limit $L$ is strictly less than $0.01$.

The population of a species of bird introduced into a nature reserve is modelled by the function $P(t) = 200 + \frac{1500t}{2t + 5}$, where $t$ is the time in years since its introduction. Calculate the long-term population limit (as $t \to +\infty$) and explain what this value means for the balance of the ecosystem.

The temperature $T$ (in degrees Celsius) of an operating electronic component $t$ minutes after being switched on follows Newton's law of cooling adapted to heating: $T(t) = 20 + 65(1 - e^{-0.1t})$. Calculate the limit of the temperature as $t \to +\infty$. What physical meaning does this value have if it is known that the component is irreversibly destroyed upon reaching $90^\circ C$?

The total cost of producing $x$ units of an advanced technological device is given by the function $C(x) = 500\,000 + 3\,000x + 1x^2$ pesos. The average cost per unit is defined as $C_m(x) = \frac{C(x)}{x}$. Calculate $\lim_{x \to +\infty} C_m(x)$ and explain, in economic terms, why this mathematical model ceases to be realistic for extremely high production levels.