Understanding Limits and Function Behavior

The video tutorial explores the function f(x) = (x^2 + x - 6) / (x - 2) and investigates the limit as x approaches 2. Initially, f(2) is evaluated, revealing the function is undefined at this point. The function is then simplified by factoring, leading to a clearer understanding of its behavior. Graphing the function helps visualize its properties, and the limit is determined both graphically and numerically, concluding that the limit is 5.

What is the initial problem with evaluating f(2) for the given function?

• The function is not defined at x = 2.

• The function is continuous at x = 2.

• The function has a maximum at x = 2.

• The function is zero at x = 2.

How can the function be simplified to find the limit?

• By factoring the numerator and canceling terms.

• By adding a constant to the numerator.

• By multiplying the denominator by x.

• By integrating the function.

What is the new expression for f(x) after simplification?

• f(x) = x^2 + 3

• f(x) = x + 3

• f(x) = x^2 - 3

• f(x) = x - 3

What is the y-intercept of the graph of the simplified function?

• 3

• 4

• 2

• 1

What happens to the function at x = 2 on the graph?

• It is continuous.

• It is undefined.

• It has a minimum.

• It has a maximum.

As x approaches 2 from the left, what value does f(x) approach?

• 3

• 4

• 5

• 6

As x approaches 2 from the right, what value does f(x) approach?

• 5

• 3

• 6

• 4

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

• 6

• 5

• 4

• 3

How does the numerical approach confirm the limit?

• By showing f(x) approaches 3.

• By showing f(x) approaches 4.

• By showing f(x) approaches 5.

• By showing f(x) approaches 6.

What happens to f(x) as x gets closer to 2 from both directions?

• It diverges.

• It approaches 5.

• It oscillates.

• It becomes undefined.