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 knowing 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 being administered 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:

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

Resolve the following limit which presents an indeterminate form of 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 with 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 rise to an indeterminate form of 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 bird species introduced into a nature reserve is modeled by the function $P(t) = 200 + \frac{1500t}{2t + 5}$, where $t$ is the time in years since its introduction. Calculate the long-term limit of the population (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:

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

Observe 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 combining exponential and polynomial terms. Take into account which terms grow faster as $x \to +\infty$:

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

Calculate the following limit as $x \to -\infty$, paying special attention to the sign of the variable when extracting it from 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 $\infty - \infty$, results in $4$:

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

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

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

The temperature $T$ (in degrees Fahrenheit) of an operating electronic component after $t$ minutes of 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 function $f(x)$, which is equal to $\frac{3x^2 - 1}{x^2 + 2}$ if $x \le 0$, and is equal to $\frac{2e^x + 5}{e^x - 1}$ 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 by applying 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 $\$C(x) = 25000 + 150x + 0.05x^2$. The average cost per unit is defined as $C_m(x) = \frac{C(x)}{x}$. Calculate $\lim_{x \to +\infty} C_m(x)$ and reason, 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 solve 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 rationalization and a careful analysis of the signs when dealing with the implicit modulus in the square root:

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