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Limits

A limit describes the value an expression gets closer and closer to, as x approaches a target value.

Getting Closer and Closer

Some expressions are undefined at one exact point — often because plugging that value in gives 0 \div 0 . A limit asks: ignoring that one point, what value is the expression heading towards nearby?

The expression is undefined exactly at x = 2, but from both sides it is clearly heading to 4 . So lim(x \to 2) (x^{2} - 4) \div (x - 2) = 4 .

Example 1 — finding a limit by factorising

Find lim(x \to 3) (x^{2} - 9) \div (x - 3). 1 Plugging in x = 3 gives 0 \div 0 — undefined. Factorise instead. 2 x^{2} - 9 = (x - 3)(x + 3), so the expression becomes (x - 3)(x + 3) \div (x - 3) 3 Cancel the matching (x - 3) factor (valid since x is only *approaching* 3, never equal to it): the expression simplifies to x + 3 4 Now substitute x = 3 : 3 + 3 = 6 Answer: 6

Practice

Find lim(x \to 5) (x^{2} - 25) \div (x - 5).

Your answer

Getting Closer and Closer

Some expressions are undefined at one exact point — often because plugging that value in gives 0 \div 0 . A limit asks: ignoring that one point, what value is the expression heading towards nearby?

The expression is undefined exactly at x = 2, but from both sides it is clearly heading to 4 . So lim(x \to 2) (x^{2} - 4) \div (x - 2) = 4 .

Example 1 — finding a limit by factorising

Find lim(x \to 3) (x^{2} - 9) \div (x - 3). 1 Plugging in x = 3 gives 0 \div 0 — undefined. Factorise instead. 2 x^{2} - 9 = (x - 3)(x + 3), so the expression becomes (x - 3)(x + 3) \div (x - 3) 3 Cancel the matching (x - 3) factor (valid since x is only *approaching* 3, never equal to it): the expression simplifies to x + 3 4 Now substitute x = 3 : 3 + 3 = 6 Answer: 6

Practice

Find lim(x \to 5) (x^{2} - 25) \div (x - 5).

Your answer