Q1. Evaluate the limits analytically, or if a limit does not exist, explain.

Background

Topic: Limits and Continuity

This question tests your understanding of how to evaluate limits using algebraic techniques and recognize when a limit does not exist.

Key Terms and Formulas:

• Limit:

• Direct Substitution: Plugging the value of into the function

• Indeterminate Forms: , , etc.

• Factoring, Rationalizing, and Simplifying: Techniques to resolve indeterminate forms

Step-by-Step Guidance

1. Identify the type of limit (e.g., direct substitution, indeterminate form, one-sided limit).

2. Attempt direct substitution for each limit. If you get an indeterminate form, note it.

3. If indeterminate, use algebraic techniques such as factoring or rationalizing to simplify the expression.

4. Re-evaluate the limit after simplification. If the limit still does not exist, consider whether it diverges or approaches infinity.

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Q2. Determine the following. If a limit does not exist, explain why.

Background

Topic: One-Sided Limits and Discontinuity

This question tests your ability to evaluate one-sided limits and recognize discontinuities in piecewise functions.

Key Terms and Formulas:

• One-sided limit: and

• Piecewise function: Function defined by different expressions for different intervals

• Discontinuity: Point where the function is not continuous

Step-by-Step Guidance

1. Identify the point of interest and the relevant piece of the function for each side.

2. Evaluate the left-hand and right-hand limits separately using the appropriate expressions.

3. Compare the two one-sided limits. If they are equal, the limit exists; if not, explain why.

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Q3. Using the graph of f(x) shown to the right, complete each statement.

Background

Topic: Graphical Analysis of Limits

This question tests your ability to interpret limits and function values from a graph.

Key Terms and Formulas:

• Limit from a graph: is the value the function approaches as gets close to .

• Function value: is the actual value at .

• Discontinuity: Jump, hole, or asymptote in the graph

Step-by-Step Guidance

1. Locate the point on the graph.

2. Observe the behavior of as approaches from both sides.

3. Determine if the left and right limits are equal and if the function value matches the limit.

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Q4. Use the Squeeze Theorem. Given satisfies and , find .

Background

Topic: Squeeze Theorem

This question tests your understanding of how to use the Squeeze Theorem to evaluate limits.

Key Terms and Formulas:

• Squeeze Theorem: If and , then .

Step-by-Step Guidance

1. Identify the functions , , and and the interval where the inequalities hold.

2. Evaluate the limits of and as approaches .

3. Confirm that both limits are equal to .

4. Conclude that must also be by the Squeeze Theorem.

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Q5. Find vertical and horizontal asymptotes and left- and right-sided limits at the asymptotes.

Background

Topic: Asymptotes and Limits

This question tests your ability to find vertical and horizontal asymptotes and evaluate limits at those points.

Key Terms and Formulas:

• Vertical asymptote: where or diverges

• Horizontal asymptote: where

Step-by-Step Guidance

1. Identify points where the denominator of is zero (potential vertical asymptotes).

2. Evaluate left- and right-sided limits at these points to confirm vertical asymptotes.

3. Analyze the behavior of as and to find horizontal asymptotes.

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Q6. Find limits using asymptotes and show the limit as and . If you need to use the Squeeze Theorem, explain why.

Background

Topic: Limits at Infinity and Squeeze Theorem

This question tests your ability to evaluate limits as approaches infinity and recognize when the Squeeze Theorem is applicable.

Key Terms and Formulas:

• Limit at infinity:

• Squeeze Theorem: Used when is bounded between two functions with known limits

Step-by-Step Guidance

1. Analyze the degree of the numerator and denominator to determine the behavior as .

2. If is bounded, identify bounding functions and check their limits.

3. Explain why the Squeeze Theorem applies if used.

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Q7. Find the x values at which f is discontinuous and classify each as removable or non-removable.

Background

Topic: Discontinuity and Classification

This question tests your ability to identify points of discontinuity and classify them as removable or non-removable.

Key Terms and Formulas:

• Removable discontinuity: Can be "fixed" by redefining at a point

• Non-removable discontinuity: Cannot be fixed, often due to jump or infinite discontinuity

Step-by-Step Guidance

1. Find points where the function is not defined or the limit does not match the function value.

2. Check if the limit exists at those points. If so, the discontinuity is removable; if not, it is non-removable.

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Q8. Apply the Intermediate Value Theorem to determine if the function f(x) is continuous on the interval and if it takes on a given value.

Background

Topic: Intermediate Value Theorem

This question tests your understanding of the Intermediate Value Theorem and continuity on an interval.

Key Terms and Formulas:

• Intermediate Value Theorem: If is continuous on and is between and , then for some in .

• Continuity: is continuous if for all in the interval

Step-by-Step Guidance

1. Check if is continuous on the given interval.

2. Evaluate and and determine if the target value lies between them.

3. Apply the theorem to conclude if takes on the value .

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Q9. List all the conditions that must be true for a function to be continuous at a point.

Background

Topic: Continuity at a Point

This question tests your understanding of the formal definition of continuity at a point.

Key Terms and Formulas:

• Continuity at requires:

  • is defined

  • exists

Step-by-Step Guidance

1. State each condition clearly and explain why it is necessary for continuity.

2. Provide examples or counterexamples for each condition.

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Q10. Use the graph to determine the largest value of δ so that whenever for .

Background

Topic: Precise Definition of a Limit (Epsilon-Delta)

This question tests your understanding of the epsilon-delta definition of a limit using a graph.

Key Terms and Formulas:

• Epsilon (): Desired closeness to the limit value

• Delta (): Corresponding closeness to the input value

• For , whenever

Step-by-Step Guidance

1. Set up the inequality and relate it to .

2. Use the graph to visually estimate the largest possible that satisfies the condition.

3. Check your estimate by plugging values into the inequality.

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