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 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

Compare degrees of numerator and denominator. They are equal (degree 2). Divide leading coefficients: 3/1 = 3. HA is y = 3.

Recognize exponential growth is faster than polynomial. The limit is 0.

Compare the degrees of the numerator and denominator. Apply the 'bottom heavy', 'top heavy', or 'equal degree' rules.

Check if the oscillating function is divided by a function that approaches infinity. If so, the Squeeze Theorem often applies, resulting in a limit of 0. Otherwise, the limit DNE.

Divide each term by x: $\lim_{x \to \infty} (4 + \frac{\sin(x)}{x})$. Since $\lim_{x \to \infty} \frac{\sin(x)}{x} = 0$, the limit is 4.

Consider the leading term of the polynomial. If the degree is even, both ends go to +∞ or -∞ (depending on the sign of the leading coefficient). If the degree is odd, one end goes to +∞ and the other to -∞.

Recognize that polynomial growth is faster than logarithmic. The limit is 0.

Set the denominator equal to zero and solve for x. These x-values are potential vertical asymptotes.

As x approaches negative infinity, $e^x$ approaches 0.

Solve for $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$.