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 litre, $t$ 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 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 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:

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