Video Notes: Limits, Continuity, Differentiation, and Chain Rule

Flashcards covering key concepts from Limits & Continuity, Differentiation, and the Chain Rule (Units 1–3).

What is the limit of a function as x approaches a point c?

The value that f(x) gets arbitrarily close to as x approaches c (x ≠ c), if it exists.

How do you find limits from a table?

Examine values of f(x) as x approaches the target value from the table and observe the trend toward a limiting value.

How do you find limits from a graph?

Identify the y-value approached by the graph as x approaches the target value.

What are one-sided limits?

Limits as x approaches c from the left (x → c−) or from the right (x → c+).

What are the basic limit properties (including composition)?

Limit laws include sums, differences, products, quotients, and constants; the limit of a composition uses the outer function evaluated at the inner limit when appropriate.

How do you evaluate a limit algebraically when you get the indeterminate form 0/0?

Factor, cancel, rearrange, or simplify until the limit can be found; substitute afterwards.

What are the types of discontinuities?

Removable, jump, and infinite (essential) discontinuities.

What does it mean for a function to be continuous at a point a?

The limit as x approaches a of f(x) equals f(a) (f(a) must exist).

What are vertical and horizontal asymptotes?

Vertical: x = c where f(x) grows without bound as x approaches c. Horizontal: y = L as x approaches ±∞.

What are limits to infinity?

The behavior of f(x) as x grows without bound (x → ±∞); the limit may be a finite value or ±∞.

What is the Intermediate Value Theorem?

If f is continuous on [a,b] and k lies between f(a) and f(b), then there exists a c in (a,b) with f(c) = k.

What is a derivative?

The instantaneous rate of change; the slope of the tangent line; defined as the limit of the average rate of change as the interval shrinks.

Is the derivative a function?

Yes, it is a function f′(x) that gives the slope of f at each x where defined.

How are average and instantaneous rates of change related?

Instantaneous rate is the limit of the average rate as the interval tends to zero.

What is the equation of the tangent line to y = f(x) at x = a?

y = f(a) + f′(a)(x − a).

Power Rule

d/dx [x^n] = n x^{n-1}.

Product Rule

If y = u v, then y′ = u′ v + u v′.

Quotient Rule

If y = u/v, then y′ = (u′ v − u v′)/v^2.

Derivatives of the six trigonometric functions

sin′x = cos x; cos′x = −sin x; tan′x = sec^2 x; cot′x = −csc^2 x; sec′x = sec x tan x; csc′x = −csc x cot x.

Derivatives of e^x and ln x

d/dx e^x = e^x; d/dx ln x = 1/x (x > 0).

Finding derivatives using multiple representations

Derivatives can be found analytically using rules or by interpreting values from tables/graphs as needed.

What does it mean to be differentiable?

A function is differentiable at a if f′(a) exists; differentiability implies continuity.

What is the Chain Rule?

If y = f(g(x)), then dy/dx = f′(g(x)) · g′(x).