Understanding Limits of Rational Functions

The video tutorial explains how to determine the limit of a rational function as x approaches 3. It begins with an analytical approach, highlighting the issue of division by zero when using direct substitution. The function is then simplified by factoring, allowing the limit to be found. The tutorial concludes with a graphical verification, showing a hole at x = 3 and confirming the limit is 1/2.

What is the initial method suggested for finding the limit of the rational function as x approaches 3?

• Using a calculator

• Numerical approximation

• Direct substitution

• Graphical analysis

Why does direct substitution initially fail when finding the limit as x approaches 3?

• The function has a maximum at x = 3

• The function is continuous at x = 3

• The function is undefined at x = 3

• The function is linear at x = 3

What is the first step in simplifying the rational function?

• Adding a constant

• Factoring the numerator and denominator

• Multiplying by a conjugate

• Using the quadratic formula

What common factor is identified in both the numerator and denominator during simplification?

• X - 5

• X + 3

• X - 3

• X + 5

After simplifying the function, what expression is used to find the limit?

• 4 divided by (X - 5)

• 4 divided by (X + 3)

• 4 divided by (X + 5)

• 4 divided by (X - 3)

What is the limit of the function as x approaches 3 after simplification?

• 2/3

• 1/2

• 3/4

• 1/3

What graphical feature is present at x = 3 in the original function?

• A vertical asymptote

• A hole

• A maximum point

• A minimum point

What does the presence of a hole at x = 3 indicate about the function?

• The function is continuous at x = 3

• The function is undefined at x = 3

• The function has a maximum at x = 3

• The function has a minimum at x = 3

What other graphical feature is mentioned in the video that does not affect the limit?

• A horizontal asymptote at x = 3

• A maximum point at x = 3

• A hole at x = -5

• A vertical asymptote at x = -5

What is the significance of the function values approaching 1/2 as x approaches 3?

• The function has a minimum at 1/2

• The function has a maximum at 1/2

• The limit is 1/2 even though the function is undefined at x = 3

• The function equals 1/2 at x = 3