Limits at Infinity and Asymptotes: Fresh Take

Lumen Learning

Limits at Infinity and Asymptotes: Fresh Take

Determine limits and predict how functions behave as x increases or decreases indefinitely
Identify and distinguish horizontal and slanting lines that a graph approaches but never touches
Use a function’s derivatives to accurately sketch its graph

Limits at Infinity

The Main Idea

Limits at Infinity:
- Describes behavior of $f(x)$ as $x$ approaches $\infty$ or $-\infty$
- Notation: $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$
Horizontal Asymptotes:
- Line $y = L$ where $f(x)$ approaches $L$ as $x \to \pm\infty$
- Function may cross horizontal asymptote multiple times
Infinite Limits at Infinity:
- When $f(x)$ grows without bound as $x \to \pm\infty$
- Notation: $\lim_{x \to \infty} f(x) = \infty$ or $\lim_{x \to \infty} f(x) = -\infty$
Formal Definitions:
- For finite limits: $\forall \varepsilon > 0, \exists N > 0: |f(x) - L| < \varepsilon$ when $x > N$
- For infinite limits: $\forall M > 0, \exists N > 0: f(x) > M$ when $x > N$
Strategies for Evaluation:
- Use algebraic limit laws
- Apply the squeeze theorem
- Analyze rational functions by comparing highest degree terms

Evaluate $\underset{x\to −\infty}{\lim}\left(3+\frac{4}{x}\right)$ and $\underset{x\to \infty }{\lim}\left(3+\frac{4}{x}\right)$ . Determine the horizontal asymptotes of $f(x)=3+\frac{4}{x}$ , if any.

$\underset{x\to \pm \infty }{\lim}1/x=0$

Both limits are $3$ . The line $y=3$ is a horizontal asymptote.

Use the formal definition of limit at infinity to prove that $\underset{x\to \infty }{\lim}\left(3 - \dfrac{1}{x^2}\right)=3$ .

Let $N=\dfrac{1}{\sqrt{\varepsilon }}$ .

Let $\varepsilon >0$ . Let $N=\dfrac{1}{\sqrt{\varepsilon }}$ . Therefore, for all $x>N$ , we have

$|3-\dfrac{1}{x^2}-3|=\dfrac{1}{x^2}<\dfrac{1}{N^2}=\varepsilon$

Therefore, $\underset{x\to \infty }{\lim}\left(3-\frac{1}{x^2}\right)=3$ .

Use the formal definition of infinite limit at infinity to prove that $\underset{x\to \infty }{\lim}3x^2=\infty$ .

Let $N=\sqrt{\frac{M}{3}}$ .

Let $M>0$ . Let $N=\sqrt{\frac{M}{3}}$ . Then, for all $x>N$ , we have,

$3x^2>3N^2=3(\sqrt{\frac{M}{3}})^2=\frac{3M}{3}=M$

End Behavior

The Main Idea

End Behavior:
- Describes how a function behaves as $x \to \pm\infty$
- Three possible outcomes: approach a finite value, approach infinity, or oscillate
Polynomial End Behavior:
- Determined by the highest degree term
- Even degree: same behavior at both ends
- Odd degree: opposite behavior at each end
Rational Function End Behavior:
- Depends on the degree relationship between numerator and denominator
- Three cases: horizontal asymptote, zero asymptote, or unbounded growth
Horizontal Asymptotes:
- $y = L$ where $\lim_{x \to \pm\infty} f(x) = L$
- For rational functions p(x)q(x):
  - If deg(p) < deg(q): $y = 0$
  - If deg(p) = deg(q): $y = \frac{a_n}{b_m}$ (ratio of leading coefficients)
  - If deg(p) > deg(q): no horizontal asymptote
Transcendental Function End Behavior:
- Trigonometric: oscillate (no limit)
- Exponential ( $e^x$ ): $\to 0$ as $x \to -\infty$ , $\to \infty$ as $x \to \infty$
- Natural log: $\to -\infty$ as $x \to 0^+$ , $\to \infty$ as $x \to \infty$

Let $f(x)=-3x^4$ . Find $\underset{x\to \infty }{\lim}f(x)$ .

The coefficient $-3$ is negative.

$−\infty$

Evaluate $\underset{x\to \pm \infty }{\lim}\dfrac{3x^2+2x-1}{5x^2-4x+7}$ and use these limits to determine the end behavior of $f(x)=\dfrac{3x^2+2x-1}{5x^2-4x+7}$ .

Divide the numerator and denominator by $x^2$ .

$\frac{3}{5}$

Evaluate $\underset{x\to \infty }{\lim}\frac{\sqrt{3x^2+4}}{x+6}$ .

Divide the numerator and denominator by $|x|$ .

$\pm \sqrt{3}$

Find the limits as $x\to \infty$ and $x\to −\infty$ for $f(x)=\dfrac{(3e^x-4)}{(5e^x+2)}$ .

$\underset{x\to \infty }{\lim}e^x=\infty$ and $\underset{x\to -\infty }{\lim}e^x=0$ .

$\underset{x\to \infty }{\lim}f(x)=\frac{3}{5}$ , $\underset{x\to −\infty }{\lim}f(x)=-2$

Drawing Graphs of Functions

Sketch a graph of $f(x)=(x-1)^3 (x+2)$

$f$ is a fourth-degree polynomial.

Sketch a graph of $f(x)=\dfrac{3x+5}{8+4x}$

A line $y=L$ is a horizontal asymptote of $f$ if the limit as $x\to \infty$ or the limit as $x\to −\infty$ of $f(x)$ is $L$ . A line $x=a$ is a vertical asymptote if at least one of the one-sided limits of $f$ as $x\to a$ is $\infty$ or $−\infty$ .

Find the oblique asymptote for $f(x)=\dfrac{3x^3-2x+1}{2x^2-4}$

$y=\frac{3}{2}x$

Consider the function $f(x)=5-x^{\frac{2}{3}}$ . Determine the point on the graph where a cusp is located. Determine the end behavior of $f$ .

A function $f$ has a cusp at a point $a$ if $f(a)$ exists, $f^{\prime}(a)$ is undefined, one of the one-sided limits as $x\to a$ of $f^{\prime}(x)$ is $+\infty$ , and the other one-sided limit is $−\infty$ .

The function $f$ has a cusp at $(0,5)$ : $\underset{x\to 0^-}{\lim}f^{\prime}(x)=\infty$ , $\underset{x\to 0^+}{\lim}f^{\prime}(x)=−\infty$ . For end behavior, $\underset{x\to \pm \infty }{\lim}f(x)=−\infty$ .