Limits Involving Infinity
- Calculus 2.4
Limits by approaching infinity
We say that $f(x)$ has the limit $L$ as $x$ approaches infinity and write
\[ \lim_{x \to \infty} f(x) = L \]
if, for every number $\epsilon > 0$, there exists a number $M$ such that
\[ x > M \implies |f(x) - L| < \epsilon \]
Similarly, we say that $f(x)$ has the limit $L$ as $x$ approaches negative infinity and write
\[ \lim_{x \to -\infty} f(x) = L \]
if, for every number $\epsilon > 0$, there exists a number $N$ such that \[ x < N \implies |f(x) - L| < \epsilon \]
Limit Laws
All the limit laws here are true when $\lim_{x\to a}$ replaced to $\lim_{x\to\pm\infty}$.
Horizontal Asymptotes
A graph of a function $y=f(x)$ has a horizontal asymptote at $y = b$ if either
\[ \lim_{x \to \infty} f(x) = b \quad \text{or} \quad \lim_{x \to -\infty} f(x) = b \]
Oblique Asymptotes
If the degree of the numerator of a rational function is 1 greater than the degree of the denominator, then the graph has an oblique asymptote. The oblique asymptote can be found by performing polynomial long division on the rational function.
Infinite Limits
We say that $f(x)$ approaches infinity as $x$ approaches $a$ and write
\[ \lim_{x \to a} f(x) = \infty \]
if, for every number $M$, there exists a number $\delta > 0$ such that
\[ 0 < |x - a| < \delta \implies f(x) > M \]
Similarly, we say that $f(x)$ approaches negative infinity as $x$ approaches $a$ and write
\[ \lim_{x \to a} f(x) = -\infty \]
if, for every number $N$, there exists a number $\delta > 0$ such that
\[ 0 < |x - a| < \delta \implies f(x) < N \]
Vertical Asymptotes
A graph of a function $y=f(x)$ has a vertical asymptote at $x = a$ if either
\[ \lim_{x \to a^+} f(x) = \pm\infty \quad \text{or} \quad \lim_{x \to a^-} f(x) = \pm\infty \]
Infinite Limits at Infinity and Negative Infinity
We say that $f(x)$ approaches infinity as $x$ approaches infinity and write
\[ \lim_{x \to \infty} f(x) = \infty \]
if, for every number $M$, there exists a number $N$ such that
\[ x > N \implies f(x) > M \]
Similarly, we say that $f(x)$ approaches negative infinity as $x$ approaches infinity and write
\[ \lim_{x \to \infty} f(x) = -\infty \]
if, for every number $M$, there exists a number $N$ such that
\[ x > N \implies f(x) < N \]
We say that $f(x)$ approaches infinity as $x$ approaches negative infinity and write
\[ \lim_{x \to -\infty} f(x) = \infty \]
if, for every number $M$, there exists a number $N$ such that \[ x < N \implies f(x) > M \]
Similarly, we say that $f(x)$ approaches negative infinity as $x$ approaches negative infinity and write
\[ \lim_{x \to -\infty} f(x) = -\infty \]
if, for every number $M$, there exists a number $N$ such that \[ x < N \implies f(x) < M \]
Dominant Term
When evaluating limits at infinity, we can often simplify the function by focusing on the dominant term. The dominant term is the term in the function that grows the fastest as $x$ approaches some value or infinity. For example, consider the following function:
\[ f(x) = \frac{x^2-3}{2x-4} = \frac{x}{2} + 1 + \frac{1}{2x-4} \]
This tells us:
\[ \begin{align*} f(x) &\approx \frac{x}{2} + 1 \quad ; x \to \pm\infty \nl f(x) &\approx \frac{1}{2x-4} \quad ; x \to 2 \end{align*} \]
