AP Calculus Section 1.4: Finding Limits Numerically

Open Ended

What challenges might arise when using tables to estimate limits, and how can understanding these challenges improve your mathematical reasoning?

Multiple Choice

Why is it important to be able to estimate limit values from tables in calculus?

• Because it helps solve real-world problems where exact values are hard to find

• Because tables are easier to use than graphs

• Because it is required for all calculus exams

• Because it avoids the need for algebra

Multiple Choice

Why do we use a table to estimate limits when direct substitution fails?

• Because tables help us find the exact value at x = c

• Because tables show what f(x) approaches as x gets close to c

• Because tables are easier to use than graphs

• Because tables always give a defined value

Open Ended

Describe the process of building a table of values to estimate a limit. What are the key steps and why is it important not to include x = c itself?

Multiple Choice

What is the limit of f(x) = (x^2 - 4)/(x - 2) as x approaches 2, based on the table and graph shown?

• 0

• 2

• 4

• Undefined

Multiple Select

Which of the following statements are true about the famous limit lim(x→0) sin(x)/x?

• The limit is 0

• The limit is 1

• Both sides stabilize at the same value

• The limit does not exist

Multiple Choice

Which of the following is NOT a limitation of numerical estimation when using tables to estimate limits?

• Rounding errors

• Slow convergence

• Missing behavior

• Exact proof

Open Ended

Explain how the four representations of a limit (graphical, numerical, symbolic, and verbal) complement each other in understanding limits.

Open Ended

What does the table suggest about lim(x→2) k(x)?

Multiple Select

Select all statements that correctly describe how to estimate limit values from tables.

• Choose x-values approaching c from both sides, but never include x = c itself.

• If both sides stabilize to the same value, the two-sided limit equals that value.

• If left and right sides stabilize at different values, the limit exists.

• A table provides numerical evidence, but algebraic or analytic verification is required.

Open Ended

Reflecting on today's lesson about estimating limit values from tables, what is one concept you found most interesting or would like to explore further?