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Limit at Infinity Calculator

Evaluate limits as x → ∞ or x → -∞ to find horizontal asymptotes and analyze end behavior.

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Classic Sinc Function

Evaluate lim sin(x)/x as x approaches 0 - a foundational calculus limit

Key values: sin(x)/x · x -> 0 · Result: 1

Exponential Growth

Explore the limit of (1 + 1/x)^x as x approaches infinity (Euler's number)

Key values: (1+1/x)^x · x -> Infinity · Result: e

Difference Quotient

Evaluate the derivative definition limit for x² at x = 3

Key values: (x² - 9)/(x - 3) · x -> 3 · Result: 6

Documentation

Limits at Infinity

Evaluating the behavior of $f(x)$ as $x \to \pm\infty$ reveals the function's end behavior and identifies horizontal asymptotes.

\lim_{x \to \infty} f(x) = L \implies y = L \text{ is a horizontal asymptote}

A function can have different horizontal asymptotes as $x \to +\infty$ and $x \to -\infty$ .

Rational Functions at Infinity

For $f(x) = P(x)/Q(x)$ where $P$ has degree $m$ and $Q$ has degree $n$ :

$m < n$ : limit is $0$ (horizontal asymptote at $y = 0$ )
$m = n$ : limit is the ratio of leading coefficients
$m > n$ : limit is $\pm\infty$ (no horizontal asymptote)

Dominant term technique: Divide numerator and denominator by the highest power of $x$ in the denominator. All terms with $x$ in the denominator go to zero as $x \to \infty$ .

Common Limits at Infinity

\lim_{x \to \infty} e^{-x} = 0, \qquad \lim_{x \to \infty} \frac{\ln x}{x} = 0

\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e, \qquad \lim_{x \to \infty} \frac{\sin x}{x} = 0

The growth hierarchy: logarithms grow slower than polynomials, which grow slower than exponentials. Formally: $\ln x \ll x^a \ll b^x$ for any $a > 0$ and $b > 1$ .

Frequently Asked Questions

What is a limit at infinity?

A limit at infinity describes the behavior of $f(x)$ as $x$ grows without bound (toward positive or negative infinity). If the limit equals a finite value $L$ , then $y = L$ is a horizontal asymptote of the function.

How do I evaluate limits at infinity for rational functions?

Compare the degrees of the numerator ( $m$ ) and denominator ( $n$ ). If $m < n$ , the limit is 0. If $m = n$ , the limit is the ratio of leading coefficients. If $m > n$ , the limit is $\pm\infty$ and there is no horizontal asymptote.

What is the dominant term technique?

Divide the numerator and denominator by the highest power of $x$ in the denominator. As $x \to \infty$ , all terms with $x$ in the denominator approach zero, leaving only the dominant terms that determine the limit.

Can a function have different horizontal asymptotes in each direction?

Yes. A function can approach different values as $x \to +\infty$ versus $x \to -\infty$ . For example, $f(x) = \frac{x}{\sqrt{x^2 + 1}}$ approaches 1 as $x \to +\infty$ and $-1$ as $x \to -\infty$ .

What is the growth hierarchy for comparing functions at infinity?

Logarithms grow slower than any positive power of $x$ , which grows slower than exponential functions. Formally: $\ln x \ll x^a \ll b^x$ for any $a > 0$ and $b > 1$ . This hierarchy determines which term dominates at infinity.

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