Grade: Grade 12 Subject: Mathematics Unit: Precalculus Completion Lesson: 5 of 6 SAT: AdvancedMath ACT: Math

Common Mistakes

Identify, understand, and avoid the most frequent errors students make in precalculus problems.

Learn

Learning from common mistakes is one of the most effective ways to improve your mathematical understanding. This lesson examines typical errors in limits, sequences, and series, explaining why they occur and how to avoid them.

Why Mistakes Happen

  • Rushing: Moving too quickly without checking work
  • Misremembering Formulas: Confusing similar formulas or forgetting conditions
  • Algebraic Errors: Sign mistakes, distribution errors, and arithmetic slips
  • Conceptual Confusion: Misunderstanding when techniques apply
  • Notation Errors: Misinterpreting or miswriting mathematical symbols

Categories of Common Mistakes

  1. Limit evaluation errors
  2. Sequence formula confusion
  3. Series convergence mistakes
  4. Application and word problem errors

Examples

Study these common mistakes and their corrections carefully.

Mistake 1: Concluding a Limit is Undefined at 0/0

Wrong Approach: lim(x->2) [(x^2-4)/(x-2)] = 0/0, so the limit does not exist.

Why It's Wrong: 0/0 is an indeterminate form, not an answer. It signals that algebraic manipulation is needed.

Correct Approach: Factor: [(x+2)(x-2)]/(x-2) = x+2 for x != 2. Then lim(x->2) (x+2) = 4.

Remember: 0/0 means "keep working," not "undefined."

Mistake 2: Confusing Arithmetic and Geometric Sequences

Wrong Approach: For sequence 3, 6, 12, 24, ... using a_n = a_1 + (n-1)d

Why It's Wrong: This is a geometric sequence (each term is multiplied by 2), not arithmetic.

Correct Approach: a_n = 3 * 2^(n-1). So a_5 = 3 * 16 = 48, not 3 + 4(3) = 15.

Remember: Check if differences are constant (arithmetic) or ratios are constant (geometric).

Mistake 3: Using the Infinite Series Formula When |r| >= 1

Wrong Approach: Sum of 2 + 4 + 8 + 16 + ... = 2/(1-2) = 2/(-1) = -2

Why It's Wrong: The formula S = a/(1-r) only works when |r| < 1. Here r = 2.

Correct Approach: Since |r| = 2 > 1, this series diverges (sum approaches infinity).

Remember: Always check |r| < 1 before using the infinite series sum formula.

Mistake 4: Off-by-One Errors in Sequence Terms

Wrong Approach: For a_n = 5 + 3n, finding a_10 = 5 + 3(10) = 35

Why It's Wrong: If the formula is a_n = a_1 + (n-1)d, then a_10 = 5 + 9(3) = 32, not 35.

Correct Approach: Be clear about whether your formula is indexed from n=1 or n=0.

Remember: Verify your formula with the first few known terms before using it.

Mistake 5: Forgetting to Multiply Bouncing Ball Distance by 2

Wrong Approach: Ball dropped from 10 ft, bounces to 80%. Total = 10 + 8/(1-0.8) = 10 + 40 = 50 ft

Why It's Wrong: After the initial drop, each bounce travels UP and DOWN.

Correct Approach: Total = 10 + 2(8)/(1-0.8) = 10 + 16/0.2 = 10 + 80 = 90 ft

Remember: Bouncing objects travel the bounce distance twice (up and down).

Mistake 6: Dividing Limits Incorrectly

Wrong Approach: lim(x->inf) (3x^2 + x)/(x^2 + 1) = lim(x->inf) (3x + 1)/(x + 1) = 3

Why It's Wrong: You cannot cancel terms across addition in a fraction.

Correct Approach: Divide every term by x^2: (3 + 1/x)/(1 + 1/x^2). As x -> inf, this -> 3/1 = 3.

Remember: Divide each term by the highest power, don't cancel across addition.

Practice

Find and correct the error in each problem below.

Problem 1: A student writes: lim(x->0) [sin(x)/x] = 0/0 = 1. Is this reasoning correct? Explain.

Problem 2: Find the error: "The 20th term of 2, 5, 8, 11, ... is a_20 = 2 + 20(3) = 62."

Problem 3: Find the error: "Sum of 1 + 1/2 + 1/4 + 1/8 + ... = 1/(1 - 1/2) = 1/(1/2) = 2."

Problem 4: Find the error: "lim(x->3) [(x^2 - 9)/(x^2 - 6x + 9)] = lim(x->3) [(x-3)(x+3)/(x-3)^2] = (x+3)/(x-3) = 6/0 = undefined."

Problem 5: Find the error: "$1000 at 5% for 10 years compounded annually = 1000 * 1.05 * 10 = $10,500."

Problem 6: Find the error: "For the geometric sequence with a_1 = 4 and r = 3, the sum of the first 5 terms is S_5 = 4(1-3^5)/(1-3) = 4(-242)/(-2) = 484."

Problem 7: Find the error: "lim(x->inf) (2x + 1)/(3x - 2) = lim(x->inf) (2 + 1)/(3 - 2) = 3/1 = 3."

Problem 8: Find the error: "The sum 1 + 2 + 3 + ... + 100 = 100(100)/2 = 5000."

Problem 9: Find the error: "If a sequence has a_5 = 32 and r = 2, then a_1 = 32/2^5 = 32/32 = 1."

Problem 10: Find the error: "A ball dropped from 20 ft bounces to 1/2 its height. Total distance = 20 + 10 + 5 + ... = 20/(1-0.5) = 40 ft."

Check Your Understanding

Review the corrections for each problem.

Answer Key

  1. Incorrect reasoning. While the answer 1 is correct, "0/0 = 1" is not valid. This is a special limit that equals 1, proven by L'Hopital's rule or the squeeze theorem, not by arithmetic on 0/0.
  2. Error: Using 20 instead of 19. Correct: a_20 = 2 + (20-1)(3) = 2 + 57 = 59.
  3. This is actually correct! The sum of 1 + 1/2 + 1/4 + ... = 2. Sometimes "find the error" problems test if you can recognize correct work.
  4. Error: Evaluating after simplifying incorrectly. After canceling one (x-3), you get (x+3)/(x-3). As x->3, this approaches 6/0, which means the limit is infinity or does not exist (one-sided limits differ).
  5. Error: Using simple interest formula instead of compound. Correct: A = 1000(1.05)^10 = $1,628.89.
  6. This is correct! S_5 = 4(1-243)/(1-3) = 4(-242)/(-2) = 484.
  7. Error: Dropping the x's incorrectly. Correct: Divide by x to get (2 + 1/x)/(3 - 2/x) -> 2/3 as x -> inf.
  8. Error: Using wrong formula. Correct: S = n(a_1 + a_n)/2 = 100(1 + 100)/2 = 5050.
  9. Error: Using 2^5 instead of 2^4. a_5 = a_1 * r^4, so a_1 = 32/2^4 = 32/16 = 2.
  10. Error: Not doubling bounces. Correct: 20 + 2(10 + 5 + ...) = 20 + 2(10/(1-0.5)) = 20 + 40 = 60 ft.

Next Steps

  • Create a personal "error log" to track mistakes you commonly make
  • Before submitting any problem, check for these common errors
  • When you make a mistake, analyze why and how to prevent it
  • Proceed to the Unit Quiz to test your complete understanding