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Year 12 · Calculus

Limits

A limit is the value a function gets close to as x approaches a number.

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How to do it

  1. Try substituting the number first.
  2. If you get 0/0, factorise and cancel, then substitute.
  3. As x → ∞, divide top and bottom by the highest power of x.

Worked example

Find lim (x→∞) of (4x + 1) ÷ (3x + 5)
Divide top and bottom by x: (4 + 1/x) ÷ (3 + 5/x). As x → ∞ the 1/x and 5/x terms go to 0, leaving 4/3 = 4/3.
Answer: 4/3

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Watch out
0/0 means "do more work", not "no limit".

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