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

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) $\lim_{x \to -\infty} 4^x$

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 $x \to -\infty$.

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

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

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

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

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 $e$:

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

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.

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

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

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

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

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:

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

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

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

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 $90^\circ C$?

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

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

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.

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 $\lim_{x \to +\infty} (f(x) - mx) = n$.

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:

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

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 $0.01$.