The value that a function approaches as the input approaches some value.

A y-value that the graph of a function approaches as x tends to +∞ or -∞.

The behavior of a function as x approaches positive or negative infinity.

A function that repeatedly fluctuates between two or more values as x approaches infinity.

If g(x) ≤ f(x) ≤ h(x) for all x near c, and lim x→c g(x) = L = lim x→c h(x), then lim x→c f(x) = L.

The speed at which a function's output increases as its input increases.

A function that can be defined as a fraction where both the numerator and denominator are polynomials.

The highest power of the variable in the polynomial.

The coefficient of the term with the highest power in a polynomial.

Does Not Exist

Limits at infinity: x approaches ∞ or -∞, examining end behavior. Limits at a finite number: x approaches a specific value, examining local behavior.

Horizontal: y-value function approaches as x approaches infinity. Vertical: x-value function approaches infinity.

Polynomial: Grows at a polynomial rate (e.g., $x^2$, $x^3$). Exponential: Grows at an exponential rate (e.g., $2^x$, $e^x$). Exponential growth is much faster.

Both approach 0. $\frac{1}{x}$ can be positive or negative depending on the sign of x, while $\frac{1}{x^2}$ is always positive.

$\sin(x)$: Limit DNE (oscillates). $\frac{\sin(x)}{x}$: Limit is 0 (Squeeze Theorem).

$x^2$: As x approaches ±∞, $x^2$ approaches +∞. $x^3$: As x approaches +∞, $x^3$ approaches +∞. As x approaches -∞, $x^3$ approaches -∞.

Horizontal Asymptotes: Describe the end behavior of a function. Removable Discontinuities: Points where the function is not defined but could be redefined to be continuous.

$\ln(x)$: Approaches infinity, but very slowly. $e^x$: Approaches infinity very quickly.

Infinite Limit: The function approaches infinity as x approaches a finite value. Limit at Infinity: The function approaches a finite value or infinity as x approaches infinity.

$\frac{x+1}{x}$: Approaches 1. $\frac{x}{x+1}$: Approaches 1.

The function has a horizontal asymptote at y = 2, and $\lim_{x \to \infty} f(x) = 2$.

The function's y-values approach 0 as x goes to positive or negative infinity.

Look for a horizontal line that the graph approaches but does not cross (at least at the extreme ends).

Look for a vertical line where the function approaches infinity or negative infinity.

The function's y-values do not approach a finite limit as x goes to positive or negative infinity.

The graph approaches the x-axis (y=0) from above.

The graph increases without bound, but at a decreasing rate.

The graph oscillates with decreasing amplitude, approaching y = 0.

The function's value equals the value of the horizontal asymptote at that point. This is possible, but the function must still approach the asymptote as x goes to infinity.

$\lim_{x \to \infty} f(x) = L$ and $\lim_{x \to -\infty} f(x) = L$