Q1. Evaluate each of the following limits:

Background

Topic: Limits

This question tests your understanding of how to evaluate limits of functions as x approaches a certain value. Limits are foundational in calculus and are used to define derivatives and continuity.

Key Terms and Formulas:

• Limit: The value that a function approaches as the input approaches some value.

• Notation:

• For rational functions, check for direct substitution, factorization, or use of conjugates if you get an indeterminate form like .

Step-by-Step Guidance

1. For each limit, first try to substitute the value of x directly into the function to see if you get a defined value.

2. If direct substitution gives an indeterminate form (like ), try to simplify the function (e.g., factor numerator and denominator, rationalize, etc.).

3. If simplification is possible, cancel common factors and then substitute the value of x again.

4. If the function is not defined at the point, consider the left-hand and right-hand limits separately if needed.

5. For part (e), construct a table of values for x approaching the target value from both sides (e.g., 0.9, 0.99, 0.999, 1.001, 1.01, 1.1) and compute f(x) for each to observe the trend.

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Q2. Find the derivative at the point (x, f(x)), where f(x) = ...

Background

Topic: Derivatives

This question tests your ability to compute the derivative of a function, which represents the instantaneous rate of change or the slope of the tangent line at a point.

Key Terms and Formulas:

• Derivative:

• For polynomials:

Step-by-Step Guidance

1. Write the function f(x) explicitly (from the image in your worksheet).

2. Apply the power rule or other relevant differentiation rules to find .

3. Substitute the given value of x into to find the slope at that point.

4. If the question asks for the equation of the tangent line, use the point-slope form: .

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Q3. Let f(x) = ... (piecewise function). Evaluate the following limits and continuity:

Background

Topic: Limits and Continuity of Piecewise Functions

This question tests your understanding of how to evaluate limits from the left and right, and how to determine continuity at a point for piecewise-defined functions.

Key Terms and Formulas:

• Left-hand limit:

• Right-hand limit:

• Continuity at a point: f is continuous at x = a if

Step-by-Step Guidance

1. For each limit, identify which piece of the function applies as x approaches the given value from the left and from the right.

2. Substitute the approaching value into the appropriate piece of the function for left and right limits.

3. Compare the left-hand and right-hand limits to determine if the overall limit exists.

4. For continuity, check if the function is defined at the point and if the limit equals the function value at that point.

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Q4. Using the given graph for f, answer the following questions about limits and function values.

Background

Topic: Graphical Analysis of Limits and Function Values

This question tests your ability to interpret a graph to find limits, left-hand and right-hand limits, and function values at specific points.

Key Terms and Formulas:

• Limit from the left/right: ,

• Function value:

• Look for open and closed circles to determine function values and limits on the graph.

Step-by-Step Guidance

1. For each part, locate the relevant x-value on the graph.

2. For limits, observe the y-value the function approaches as x approaches the given value from the left and right.

3. For function values, look for a closed circle at the x-value; if there is an open circle, the function is not defined there.

4. If the left and right limits are not equal, the overall limit does not exist at that point.

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Q5. Consider the piecewise function for , for .

Background

Topic: Limits and Continuity of Piecewise Functions

This question tests your ability to evaluate limits from the left and right at the point where the definition of the function changes, and to determine continuity at that point.

Key Terms and Formulas:

• Piecewise function: A function defined by different expressions for different intervals of x.

• Left-hand limit:

• Right-hand limit:

• Continuity at x = 1:

Step-by-Step Guidance

1. For each limit, determine which piece of the function to use as x approaches 1 from the left and from the right.

2. Substitute x = 1 into the left and right pieces to find the left-hand and right-hand limits.

3. Compare the left and right limits to see if the overall limit exists at x = 1.

4. Evaluate using the appropriate piece (for ).

5. State whether the function is continuous at x = 1 by checking if the limit equals the function value.

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Q6. Determine all values of x at which the function is discontinuous.

Background

Topic: Discontinuity of Functions

This question tests your ability to identify points where a function is not continuous, which can occur due to jumps, holes, or asymptotes.

Key Terms and Formulas:

• Discontinuity: A point where the function is not continuous.

• Types of discontinuity: Removable (hole), jump, infinite (asymptote).

• Check where the function is undefined (e.g., denominator zero, piecewise breakpoints).

Step-by-Step Guidance

1. Examine the function for values of x that make the denominator zero or where the definition changes (for piecewise functions).

2. For each candidate point, check the left and right limits and the function value to determine if the function is discontinuous there.

3. List all x-values where the function fails to be continuous.

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