4.6 Limits at Infinity and Asymptotes

Limits at infinity and asymptotes describe how functions behave as $x$ gets very large (positive or negative). They tell you whether a function levels off, grows without bound, or approaches a slanted line. This is essential for sketching accurate graphs and understanding a function's long-term trend.

By evaluating these limits, you can identify horizontal and oblique asymptotes, which act as guidelines for a function's shape far from the origin.

Limits and Asymptotes

Limits at Infinity

A limit at infinity describes what happens to $f(x)$ as $x$ grows without bound in either direction.

• As $x \to \infty$, the function values may approach a finite number (a horizontal asymptote) or grow without bound.
• As $x \to -\infty$, the same possibilities apply.

Rational functions are the most common type you'll evaluate. The standard technique:

1. Identify the highest power of $x$ in the denominator.
2. Divide every term in both the numerator and denominator by that power of $x$.
3. As $x \to \infty$, any term with $x$ in the denominator goes to 0. Simplify what remains.

For example, to find $\lim_{x \to \infty} \frac{3x^2 + 1}{5x^2 - 2}$, divide everything by $x^2$:

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

Other function types:

• Exponential functions ($a^x$ where $a > 0$): If $a > 1$, then $a^x \to \infty$ as $x \to \infty$ and $a^x \to 0$ as $x \to -\infty$. If $0 < a < 1$, the behavior reverses.
• Logarithmic functions ($\ln x$): As $x \to \infty$, $\ln x \to \infty$, but very slowly compared to any positive power of $x$.
• L'Hôpital's Rule can handle indeterminate forms like $\frac{\infty}{\infty}$ or $\frac{0}{0}$ when the methods above don't resolve the limit directly.

Horizontal and Oblique Asymptotes

Horizontal asymptotes exist when $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$ for some finite constant $L$. The line $y = L$ is the asymptote.

For rational functions $\frac{P(x)}{Q(x)}$, you can determine horizontal asymptotes by comparing the degrees of the numerator and denominator:

• Degree of numerator < degree of denominator: Horizontal asymptote at $y = 0$.
• Degrees are equal: Horizontal asymptote at $y = \frac{\text{leading coefficient of } P}{\text{leading coefficient of } Q}$.
• Degree of numerator > degree of denominator: No horizontal asymptote.

Oblique (slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. To find one:

1. Perform polynomial long division of the numerator by the denominator.
2. The quotient (ignoring the remainder) gives the oblique asymptote $y = mx + b$.
3. The remainder term $\to 0$ as $x \to \infty$, so the function approaches the line $y = mx + b$.

For example, $f(x) = \frac{x^2 + 3x + 1}{x + 1}$ divides to give $x + 2$ with a remainder, so the oblique asymptote is $y = x + 2$.

Note: Exponential functions like $e^x$ do not have horizontal asymptotes as $x \to \infty$, but $e^x$ does approach 0 as $x \to -\infty$, so $y = 0$ is a horizontal asymptote in that direction.

End Behavior of Functions

End behavior describes what $f(x)$ does as $x \to \infty$ and $x \to -\infty$. For polynomials, only the leading term matters at extreme $x$-values.

For a polynomial with leading term $a_n x^n$:

• Even degree, positive $a_n$: Both ends go up ($f(x) \to \infty$ as $x \to \pm\infty$).
• Even degree, negative $a_n$: Both ends go down ($f(x) \to -\infty$ as $x \to \pm\infty$).
• Odd degree, positive $a_n$: Falls left, rises right ($f(x) \to -\infty$ as $x \to -\infty$; $f(x) \to \infty$ as $x \to \infty$).
• Odd degree, negative $a_n$: Rises left, falls right (opposite of above).

For rational functions, end behavior is captured by horizontal or oblique asymptotes. For exponential and logarithmic functions, refer to the descriptions in the section above.

Sketching Functions with Limits

Curve sketching pulls together everything from this unit. Here's the process:

1. Find the domain. Note any values of $x$ where the function is undefined.
2. Find intercepts. Set $f(x) = 0$ for $x$-intercepts; evaluate $f(0)$ for the $y$-intercept (if it exists).
3. Check for symmetry. Is $f(-x) = f(x)$ (even)? Is $f(-x) = -f(x)$ (odd)? This can cut your work in half.
4. Evaluate limits at infinity to identify horizontal or oblique asymptotes. Also check for vertical asymptotes where the denominator equals zero.
5. Find $f'(x)$ to determine where the function is increasing and decreasing. Set $f'(x) = 0$ to find critical points, then classify them as local maxima or minima.
6. Find $f''(x)$ to determine concavity. Set $f''(x) = 0$ to find possible inflection points.
7. Plot key features (intercepts, extrema, inflection points, asymptotes) and connect them, respecting the increasing/decreasing intervals and concavity you found.

Techniques for Evaluating Limits

Different limits call for different approaches. Here are the main tools, roughly in the order you should try them:

• Direct substitution: Plug in the value. If you get a finite number, you're done. This doesn't apply to limits at infinity, but it's your first move for finite limits.
• Divide by highest power of $x$: The go-to method for rational functions as $x \to \pm\infty$ (described above).
• Factoring and canceling: Useful when direct substitution gives $\frac{0}{0}$. Factor the numerator and denominator, cancel common factors, then substitute again.
• Rationalization: When you see a difference involving square roots, multiply by the conjugate. For example, to evaluate $\lim_{x \to \infty} (\sqrt{x^2 + x} - x)$, multiply and divide by $\sqrt{x^2 + x} + x$.
• L'Hôpital's Rule: When you have an indeterminate form ($\frac{0}{0}$ or $\frac{\infty}{\infty}$), take the derivative of the numerator and denominator separately, then re-evaluate the limit. You can apply it repeatedly if the result is still indeterminate.
• Graphical analysis: Use a graph to estimate or confirm your algebraic answer. This is especially helpful for checking your work on exams.