Q1. Draw a graph that satisfies all three conditions:

Background

Topic: Limits and Graphical Behavior

This question tests your understanding of how limits describe the behavior of functions near certain points and at infinity, and how to represent these behaviors graphically.

Key Terms:

• Right-hand limit: means the value approaches as approaches 1 from the right.

• Limit at infinity: describes the behavior of as becomes very large.

• Limit at negative infinity: describes the behavior of as becomes very negative.

Step-by-Step Guidance

1. Start by considering . This means as approaches 1 from the right, increases without bound. On your graph, show a vertical asymptote at where the function goes up as you approach from the right.

2. Next, tells you that as goes to negative infinity, approaches the value 3. Draw the left end of your graph approaching the horizontal line .

3. For , as goes to positive infinity, approaches 0. This means the right end of the graph should approach the horizontal line .

4. Think about what kind of function could satisfy these behaviors. For example, rational functions often have vertical and horizontal asymptotes.

5. Sketch the graph, making sure to show the vertical asymptote at , the left end approaching , and the right end approaching .

Try sketching the graph on your own before revealing the answer!

Q2. Evaluate the limits:

Background

Topic: Limits at Infinity for Rational Functions

This question tests your ability to evaluate limits as approaches infinity for rational functions, focusing on how the degrees of the numerator and denominator affect the limit.

Key Terms and Formulas:

• Degree of a polynomial: The highest power of in the expression.

• Limit at infinity: For rational functions, compare the degrees of numerator and denominator.

• General rule:

  • If degree of denominator > degree of numerator, limit is 0.

  • If degrees are equal, limit is ratio of leading coefficients.

  • If degree of numerator > denominator, limit is or .

Step-by-Step Guidance

1. For , identify the degree of numerator (1) and denominator (2).

2. Since the denominator's degree is higher, as becomes very large, the denominator grows much faster than the numerator.

3. Rewrite the expression by dividing numerator and denominator by to see the behavior as :

5. As , and both approach 0, and also approaches 0.

1. For , identify the degree of numerator (1) and denominator (1).

2. Since degrees are equal, the limit will be the ratio of leading coefficients (numerator: 2, denominator: -3).

3. Rewrite the expression by dividing numerator and denominator by :

5. As , approaches 0, so denominator approaches .

Try solving these limits on your own before revealing the answer!

Q3. Describe the continuity of .

Background

Topic: Continuity and Discontinuities of Rational Functions

This question tests your ability to analyze where a rational function is continuous and where it has discontinuities (holes or asymptotes).

Key Terms and Formulas:

• Continuous function: A function is continuous at a point if the limit exists and equals the function value.

• Discontinuity: Points where the function is not continuous. For rational functions, these occur where the denominator is zero.

• Removable discontinuity: If the numerator and denominator share a common factor, the discontinuity can be "removed" by simplifying.

• Nonremovable discontinuity: If the denominator is zero and the numerator is not, the function has a vertical asymptote.

Step-by-Step Guidance

1. Factor the numerator: .

2. Factor the denominator: .

3. Identify the values of where the denominator is zero: and .

4. Check if the numerator is also zero at these points. At , numerator is zero, so there may be a removable discontinuity (hole). At , numerator is not zero, so there is a nonremovable discontinuity (vertical asymptote).

5. Describe the continuity: The function is continuous everywhere except at (removable) and (nonremovable).

Try describing the continuity on your own before revealing the answer!

Q4. Create a function with a removable discontinuity at and a nonremovable discontinuity at .

Background

Topic: Constructing Functions with Specific Discontinuities

This question tests your ability to design a rational function with both a removable and a nonremovable discontinuity at specified points.

Key Terms and Formulas:

• Removable discontinuity: Occurs when both numerator and denominator are zero at the same value, and the factor can be cancelled.

• Nonremovable discontinuity: Occurs when only the denominator is zero at a point, resulting in a vertical asymptote.

• General form: where is removable, is nonremovable.

Step-by-Step Guidance

1. To create a removable discontinuity at , include as a factor in both numerator and denominator.

2. To create a nonremovable discontinuity at , include as a factor only in the denominator.

3. Choose a simple numerator, such as , and denominator .

4. Write the function: .

5. After simplifying, the function will have a hole at and a vertical asymptote at .

Try constructing the function on your own before revealing the answer!