1. Evaluate lim x→2 (x²−4)/(x−2), showing the factorization.
Worked answer: For x≠2, cancel (x−2) from (x−2)(x+2)/(x−2). The limit is 4.
This key corresponds by problem number to the Foundation, Core, and Extended printables. Accept mathematically equivalent justified methods. These are formative practice prompts, not secure examination items. Independent Academy assessment gates require 80% mastery.
Foundation · 12 problems
2. Evaluate lim x→3 (x+1). Explain why direct substitution works.
Worked answer: Polynomials are continuous; substitute 3, giving 4.
3. For f(x)=2x+1, calculate f(1.9), f(1.99), f(2.01), f(2.1). Predict lim x→2 f(x).
Worked answer: The values are 4.8, 4.98, 5.02, and 5.2. Both sides approach 5.
4. Evaluate lim x→0 sin(x)/x. State the unit requirement.
Worked answer: The standard trigonometric limit is 1 when x is measured in radians.
5. If lim x→1⁻ f(x)=2 and lim x→1⁺ f(x)=7, determine whether lim x→1 f(x) exists.
Worked answer: It does not exist: the one-sided limits disagree.
6. Evaluate lim x→4 (x²−16)/(x−4), for x≠4.
Worked answer: Factor the numerator to (x−4)(x+4); the limit is 8.
7. Evaluate lim x→∞ 5/x. Describe the horizontal behavior.
Worked answer: The ratio tends to 0 as x increases without bound; horizontal asymptote y=0.
8. For f(x)=(3x²+1)/(x²−4), find lim x→∞ f(x).
Worked answer: Divide numerator and denominator by x²; the limit is 3.
9. For f(x)=(x²−9)/(x−3), x≠3, identify the missing graph point.
Worked answer: For x≠3, f(x)=x+3. The graph has a removable hole at (3,6).
10. Let f(x)=x+2 when x<1 and f(x)=2x when x≥1. Does lim x→1 f(x) exist?
Worked answer: The left limit is 3 and the right limit is 2; the two-sided limit does not exist.
11. Explain whether f(a) must equal L whenever lim x→a f(x)=L. Give a counterexample.
Worked answer: No. Define f(x)=x+1 for x≠2, but f(2)=9. The limit is 3 while f(2)=9.
12. Evaluate lim x→0 |x| using left and right approach values.
Worked answer: From either side |x| approaches 0, so the limit is 0.
Core · 16 problems
1. Calculate lim x→3 (x²−9)/(x−3) and identify the excluded input.
Worked answer: Factor to x+3 for x≠3; the limit is 6. The original function excludes x=3.
2. Rationalize and evaluate lim x→4 [√(x+5)−3]/(x−4).
Worked answer: Multiply by the conjugate; obtain 1/[√(x+5)+3], giving 1/6.
3. Evaluate lim x→0 (x²−4x)/x, with x≠0.
Worked answer: Cancel x: x−4. The limit is −4.
4. Evaluate lim x→0 sin(5x)/x with x measured in radians.
Worked answer: Write 5[sin(5x)/(5x)]; the limit is 5.
5. Evaluate lim x→0 [1−cos(x)]/x using a trigonometric identity.
Worked answer: Rationalize: sin²(x)/[x(1+cos x)]. It tends to 0.
6. For f(x)=(x²−1)/(x−1) if x≠1 and f(1)=9, classify continuity at x=1.
Worked answer: The limit is 2 but f(1)=9, so the discontinuity is removable.
7. Compute lim x→−2 (x²+3x+2)/(x+2).
Worked answer: Factor (x+1)(x+2), cancel for x≠−2, then evaluate x+1=−1.
8. Find the horizontal asymptote of f(x)=(2x²+3)/(x²−1).
Worked answer: The same-degree leading coefficient ratio is 2/1; y=2.
9. Calculate lim x→∞ (3x²−x+1)/(2x²+5).
Worked answer: Divide by x²; the limit is 3/2.
10. Evaluate lim x→∞ x/(x²+1), explaining the dominant terms.
Worked answer: The denominator grows quadratically while the numerator grows linearly; limit 0.
11. Create a left/right table for f(x)=x² near x=2 and state its limit.
Worked answer: Example: f(1.9)=3.61, f(1.99)=3.9601, f(2.01)=4.0401, f(2.1)=4.41. Limit 4.
12. Evaluate lim x→0 tan(x)/x in radians.
Worked answer: tan(x)/x=[sin(x)/x]/cos(x); limit 1/1=1.
13. For f(x)=x² when x<0 and f(x)=x+1 when x≥0, evaluate lim x→0 f(x).
Worked answer: Left limit 0, right limit 1: the two-sided limit does not exist.
14. Evaluate lim x→∞ [7−3/x].
Worked answer: The term 3/x tends to 0, leaving 7.
15. Evaluate lim x→1 (x³−1)/(x−1) for x≠1.
Worked answer: Factor x³−1=(x−1)(x²+x+1); the limit is 3.
16. For f(x)=(x²−4)/(x−2) when x≠2 and f(2)=7, compare the limit and f(2).
Worked answer: The limit is 4, while the assigned function value is 7; discontinuous at x=2.
Extended · 12 problems
1. Evaluate lim x→0 [√(4+x)−2]/x and justify rationalization.
Worked answer: The conjugate gives 1/[√(4+x)+2], so the limit is 1/4.
2. Evaluate lim x→0 sin(3x)/sin(5x), in radians.
Worked answer: Write [sin(3x)/(3x)]/[sin(5x)/(5x)]×3/5, yielding 3/5.
3. Evaluate lim x→∞ (4x³−2x)/(2x³+7) with leading-term reasoning.
Worked answer: Divide by x³; all lower-order terms vanish. The limit is 2.
4. Determine lim x→∞ (x²+1)/(x+2), including its sign.
Worked answer: The ratio grows like x, so it tends to +∞. No finite horizontal asymptote as x→∞.
5. Test the existence of lim x→0 |x|/x using two one-sided calculations.
Worked answer: For x<0 the ratio is −1; for x>0 it is +1. The two-sided limit does not exist.
6. Calculate lim x→2 (x³−8)/(x−2) using a difference of cubes.
Worked answer: Cancel (x−2) in (x−2)(x²+2x+4); evaluate to 12.
7. Choose k so that f(x)=x²+k for x<1 and f(x)=3x for x≥1 is continuous at 1.
Worked answer: Match one-sided values: 1+k=3, so k=2.
8. Evaluate lim x→0 [e^(2x)−1]/x. Identify the underlying standard limit.
Worked answer: Substitute u=2x: 2(e^u−1)/u tends to 2.
9. Compute lim x→∞ √(x²+4)/x, carefully handling absolute value.
Worked answer: For positive x, √(x²+4)/x=√(1+4/x²), tending to +1.
10. Compute lim x→−∞ √(x²+4)/x. Explain the different sign.
Worked answer: Here x<0, so |x|/x=−1. The expression tends to −1.
11. Give a function continuous at zero but not differentiable there, and justify.
Worked answer: f(x)=|x| is continuous at 0. Left derivative −1 and right derivative +1 differ.
12. Give a function that crosses its own horizontal asymptote, and prove both claims.
Worked answer: f(x)=x/(x²+1) crosses y=0 at x=0, while lim x→±∞ f(x)=0.
Jennifer Kay Pearl · Khaemenes Higher Learning · Collegiate development and faculty review