Limits: watch it solved step by step

Where does a function head as x approaches a number or infinity? Direct substitution, a two-sided table and the graph. Level: A-level. Type any problem of this kind and the Lab works it out and animates every move, with a plain-words explanation of why.

Worked example: (x^2 - 1)/(x - 1) as x -> 1

1. The question. A limit asks what value f(x) gets closer and closer to as x approaches a number, even if x never actually reaches it.
2. Direct substitution fails. Putting x = 1 in gives something undefined (such as 0/0). That does not mean there is no limit: it means we must look at nearby values.
3. Squeeze in from both sides. Take values of x closer and closer to the number, from below (left) and from above (right), and see where f(x) goes.
4. Read off the limit. Both sides crowd in on the same value, so that is the limit.
5. Check with L'Hôpital's rule. When a quotient gives 0/0, the limit equals the ratio of the derivatives: differentiate the top and the bottom separately.
6. How sure are we?. A table is strong evidence, not a proof. To prove a limit you would simplify algebraically (for example cancel a common factor) or use a theorem such as L'Hôpital's rule.

Answer: 2

Practise limits with problems the Lab marks for you.