Study guides › Level 10 · Calculus

Limits

Level 10 · Calculus · for questions like “As x gets closer to 3, what does (x² − 9) ÷ (x − 3) get closer to?”

💡 The idea

A limit asks: what value does this get closer and closer to? It doesn't matter what happens exactly at that point.

🪜 The steps

1. Try putting the number in. If you get a normal number, that's the limit.
2. If you get 0 ÷ 0, simplify first. Factor and cancel, then put the number in.
3. As x gets huge, only the biggest powers matter. Compare their numbers in front.

✏️ Worked examples

1. (x² − 9) ÷ (x − 3) as x → 3: putting 3 in gives 0 ÷ 0. Factor: (x − 3)(x + 3) ÷ (x − 3) = x + 3. Then 3 + 3 = 6.

2. (6x² + 1) ÷ (2x² + 5) as x gets huge: about 6x² ÷ 2x² = 3.

3. x² + 2x + 1 as x → 2: 4 + 4 + 1 = 9.

⚠️ Watch out

• 0 ÷ 0 doesn't mean “no answer”. It means: simplify first.

🎯 Try these

1. (x² − 16) ÷ (x − 4) as x → 4
2. (10x² + 3) ÷ (5x² − 1) as x gets huge
3. x² − 3x + 5 as x → 3

Answers: 1) 8 2) 2 3) 5

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.