What is the mathematical expression for the limit of a function f(x) as x approaches a?

Limₓ → a f(x) = L

Fill in the blank: A function is said to be ______ if the derivative of the function exists at all points of its domain.

Differentiable

True or False: If a function is continuous at a point, it is always differentiable at that point.

False. A function can be continuous at a point but not differentiable there, as in the case of f(x) = |x| at x = 0.

Which theorem states that if a function is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the derivative equals the average rate of change?

Lagrange's Mean Value Theorem

Riddle: I can be found when you differentiate, my value can tell you how functions behave. What am I?

The derivative

What rule is used to evaluate limits of the form 0/0 or ∞/∞?

L'Hospital's Rule

Fill in the blank: A function f(x) is said to be ______ at a point c if f'(c) = 0 and f''(c) <>

Local maxima

True or False: The absolute maximum value of a function is always equal to the local maximum value.

False. The absolute maximum is the highest value on a closed interval, while local maxima can be lower than the global maximum.

Which condition must be satisfied for the existence of a limit at a point a?

The left-hand limit and the right-hand limit must exist and be equal.

What is the condition for a function to be continuous at a point a?

F(a) exists and is equal to both the left-hand limit and right-hand limit at a.