Understanding Limits in Calculus

The video introduces the concept of limits, a fundamental idea in calculus. It begins by defining a simple function with a discontinuity and explains how to graph it, highlighting undefined points. The concept of limits is explored as x approaches a value, using both visual and numerical methods. An example with a parabola is provided to illustrate limits with a discontinuity, and a calculator is used to demonstrate a numerical approach to understanding limits.

What is the primary concept that calculus is based upon?

• Limits

• Derivatives

• Functions

• Integrals

Why is the function f(x) = (x-1)/(x-1) undefined at x=1?

• Because it is a constant function

• Because it is a linear function

• Because it results in division by zero

• Because it is a quadratic function

How can the function f(x) = (x-1)/(x-1) be simplified?

• f(x) = 1, with x ≠ 1

• f(x) = 0

• f(x) = x - 1

• f(x) = x

What does the graph of f(x) = (x-1)/(x-1) look like for x ≠ 1?

• A sine wave

• A parabola

• A vertical line at x = 1

• A horizontal line at y = 1

What is the value of f(x) as x approaches 1?

• 1

• Undefined

• Infinity

• 0

What is the function g(x) equal to when x = 2?

• x^2

• 2

• 1

• 0

What is the limit of g(x) as x approaches 2?

• 1

• 2

• 3

• 4

How does the graph of g(x) behave at x = 2?

• It is continuous

• It has a gap

• It is a straight line

• It is undefined

What happens to g(x) as x gets closer to 2 from either direction?

• It approaches 2

• It approaches 3

• It approaches 1

• It approaches 4

What does the numerical approach to finding limits involve?

• Using a graph

• Using a ruler

• Using a calculator

• Using a compass