$\lim_{x \to a} f(x)g(x) = \pm\infty$, if $c > 0$.
$\lim_{x \to a} f(x)g(x) = \mp\infty$, if $c < 0$.

Definition (Vertical Asymptote): The line $x = a$ is a vertical asymptote of the graph of $y = f(x)$ if at least one of the following is true:

$\lim_{x \to a^-} f(x) = -\infty$
$\lim_{x \to a^-} f(x) = +\infty$
$\lim_{x \to a^+} f(x) = -\infty$
$\lim_{x \to a^+} f(x) = +\infty$

Indeterminate form of type $\infty - \infty$: If $\lim_{x \to a} f(x) = +\infty$ while $\lim_{x \to a} g(x) = -\infty$, then $\lim_{x \to a} [f(x) - g(x)]$ is called an indeterminate form of type $\infty - \infty$.

Indeterminate form of type $0 \cdot \infty$: If $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = \pm\infty$, then $\lim_{x \to a} [f(x)g(x)]$ is called an indeterminate form of type $0 \cdot \infty$.

1.3.2 Limits at Infinity

Definition (Limit to a real number L): Let $f$ be a function defined at every number in some interval $(a, \infty)$. We say that the limit of $f(x)$ as $x$ approaches positive infinity is $L$, denoted $\lim_{x \to +\infty} f(x) = L$, if the values of $f(x)$ get closer and closer to $L$ as the values of $x$ increase without bound.
Definition (Limit to a real number L from negative infinity): Similarly, let $f$ be a function defined at every number in some interval $(-\infty, a)$. We say that the limit of $f(x)$ as $x$ approaches negative infinity is $L$, denoted $\lim_{x \to -\infty} f(x) = L$, if the values of $f(x)$ get closer and closer to $L$ as the values of $x$ decrease without bound.
Remark 1.3.14: We have similar notions for the following symbols:
- $\lim_{x \to +\infty} f(x) = +\infty$ or $-\infty$
- $\lim_{x \to -\infty} f(x) = +\infty$ or $-\infty$
Theorem 1.3.16: Let $n$ be a positive integer.
1. $\lim_{x \to +\infty} x^n = +\infty$, if $n$ is even.
2. $\lim_{x \to +\infty} x^n = +\infty$, if $n$ is odd.
3. $\lim_{x \to +\infty} \frac{1}{x^n} = 0$.
4. Let $c \in \mathbb{R}$. Suppose $\lim_{x \to +\infty} f(x) = c$ and $\lim_{x \to +\infty} g(x) = \pm\infty$. Then $\lim_{x \to +\infty} \frac{f(x)}{g(x)} = 0$.
Remark 1.3.17: In statement 4 of the previous theorem, "$x \to +\infty$" may be replaced by "$x \to -\infty$", "$x \to a^-$", "$x \to a^+$", and "$x \to a$". What is important is that the limit of the numerator exists, while the denominator increases or decreases without bound.
Remark 1.3.20 (Limits of Polynomial Functions): In general, to find $\lim_{x \to \pm\infty} f(x)$ if $f$ is a polynomial function, it suffices to consider the behavior of the leading term of $f(x)$ as $x \to \pm\infty$.
Indeterminate form of type $\infty/\infty$: If $\lim_{x \to a} f(x) = \pm\infty$ and $\lim_{x \to a} g(x) = \pm\infty$, then $\lim_{x \to a} \frac{f(x)}{g(x)}$ is called an indeterminate form of type $\infty/\infty$.
Definition (Horizontal Asymptote): The line $y = L$ is a horizontal asymptote of the graph of $y = f(x)$ if $\lim_{x \to -\infty} f(x) = L$ or $\lim_{x \to +\infty} f(x) = L$.
Definition (Oblique Asymptote): The graph of $f$ has the line $y = mx + b$, $m \neq 0$, as an oblique asymptote if at least one of the following is true:
- $\lim_{x \to +\infty} [f(x) - (mx + b)] = 0$
- $\lim_{x \to -\infty} [f(x) - (mx + b)] = 0$

1.4 Limit of a Function: The Formal Definition

Definition (Formal Definition of