The slope of the secant line between two points on a function.

The slope of the tangent line at a single point on a function.

The value a function approaches as the input (x-value) gets closer to a certain point.

The limit as x approaches a value from either the left or the right.

The limit as x approaches a value from both the left and the right. Both one-sided limits must be equal for it to exist.

The function 'jumps' from one value to another.

A 'hole' in the graph that can be 'filled' by redefining the function.

A vertical asymptote where the function approaches infinity or negative infinity.

f(a) is defined, the limit of f(x) as x approaches a exists, and the limit equals f(a).

A vertical line x = a where the function approaches infinity or negative infinity as x approaches a.

1. Substitute. 2. Factor. 3. Find a Common Denominator. 4. Multiply by the Conjugate.

1. Check if f(a) is defined. 2. Check if the limit as x approaches a exists. 3. Check if the limit equals f(a).

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

1. Find two functions that 'sandwich' the given function. 2. Show that the limits of the outer functions are equal as x approaches a certain value. 3. Conclude that the limit of the inner function is the same.

1. Verify the function is continuous on the closed interval [a,b]. 2. Check if the desired value is between f(a) and f(b). 3. Conclude that there exists a c in [a,b] such that f(c) equals the desired value.

Compare the degrees of the numerator and denominator. If equal, the limit is the ratio of leading coefficients. If the denominator's degree is larger, the limit is 0.

If $g(x) \leq f(x) \leq h(x)$ for all x near a, and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

If f is continuous on [a, b] and k is any number between f(a) and f(b), then there exists at least one number c in [a, b] such that f(c) = k.