Calculus: Limits and Derivatives

Graphical limits: As x approaches 2, a graph approaches y=5 from both sides, but the plotted point is (2,1). What is the limit of f(x) as x approaches 2?

Graphical limits: A graph approaches y=3 from the left and y=7 from the right as x approaches 1. Does the limit exist?

Graphical limits: A graph approaches y=-2 from both sides as x approaches 4, with a hole at (4,-2). What is the limit of f(x) as x approaches 4?

Graphical limits: As x approaches 0, a graph rises without bound from both sides. What is the limit of f(x) as x approaches 0?

Graphical limits: A graph approaches y=0 from the left and y=2 from the right as x approaches 3. What is the left-hand limit?

Algebraic limits: Evaluate the limit of 2x+1 as x approaches 3.

Algebraic limits: Evaluate the limit of x²+3x as x approaches 2.

Algebraic limits: Evaluate the limit of (x²-4)/(x-2) as x approaches 2.

Algebraic limits: Evaluate the limit of (x²+x-2)/(x-1) as x approaches 1.

Algebraic limits: Evaluate the limit of (√(x+4)-2)/x as x approaches 0.

Definition of derivatives: What is the limit definition of f′(a)?

Definition of derivatives: Using the limit definition, what is the derivative of f(x)=x²?

Definition of derivatives: Using the limit definition, what is the derivative of f(x)=3x+2?

Definition of derivatives: Using the limit definition, find f′(2) for f(x)=x²+1.

Definition of derivatives: What does the derivative represent geometrically?

Graphical derivatives: A graph has a horizontal tangent at x=2. What is f′(2)?

Graphical derivatives: The tangent line to a graph at x=1 rises 3 units for every 2 units to the right. What is f′(1)?

Graphical derivatives: A graph is decreasing at x=4. What can be said about f′(4)?

Graphical derivatives: A graph has a sharp corner at x=0. Does f′(0) exist?

Graphical derivatives: A graph is increasing and becomes steeper as x increases. Is its derivative positive or negative, and is it increasing or decreasing?

Graphical limits: From a graph, f(x) approaches 4 as x approaches 2 from the left and 6 from the right. What is the two-sided limit?

Graphical limits: A graph has a filled point at (3,5), but its curve approaches y=1 from both sides as x approaches 3. Find the limit and f(3).

Graphical limits: A graph approaches y=-2 from the left and y=-2 from the right as x approaches 0, but f(0) is undefined. Does the limit exist?

Graphical limits: A graph rises without bound as x approaches 5 from the left and falls without bound as x approaches 5 from the right. Does the two-sided limit exist?

Graphical limits: A graph approaches y=3 from both sides as x approaches -1, and f(-1)=3. What is the limit as x approaches -1?

Algebraic limits: Evaluate the limit of (x²-9)/(x-3) as x approaches 3.

Algebraic limits: Evaluate the limit of (x²-5x+6)/(x-2) as x approaches 2.

Algebraic limits: Evaluate the limit of (x³-8)/(x-2) as x approaches 2.

Algebraic limits: Evaluate the limit of (√(x+5)-3)/(x-4) as x approaches 4.

Algebraic limits: Evaluate the limit of (1/x - 1/2)/(x-2) as x approaches 2.

Definition of derivatives: Use the limit definition to find f′(x) for f(x)=x²+2x.

Definition of derivatives: Use the limit definition to find f′(3) for f(x)=x².

Definition of derivatives: Use the limit definition to find f′(x) for f(x)=5.

Definition of derivatives: Use the limit definition to find f′(x) for f(x)=1/x, where x is not 0.

Definition of derivatives: For f(x)=x², write the difference quotient used to find f′(a).

Graphical derivatives: A tangent line at x=2 passes through (2,3) and (4,7). Find f′(2).

Graphical derivatives: A graph has a local maximum at x=1 and a smooth curve there. What is f′(1)?

Graphical derivatives: A graph has a vertical tangent at x=2. Is its derivative finite there?

Graphical derivatives: The tangent slope is -3 at x=4. Is the graph increasing or decreasing there?

Graphical derivatives: A graph is increasing but flattening as x increases. Is its derivative positive or negative, and is the derivative increasing or decreasing?