The Limit Laws and Indeterminate Forms
Combine limits with the rules for sums, products and quotients, and recognise the forms those rules leave undecided. A 0/0 quotient is a signal to factorise, rationalise or squeeze.
What a learner can do afterwards
- Evaluate a limit by cancelling a removable factor
- Clear a 0/0 form in a surd expression by rationalising
- Trap a function between two others to force a limit by the squeeze theorem
1 · Read
Limit laws split one scary limit into small friendly ones. The limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and the limit of a quotient is the quotient of the limits, provided the bottom limit is not zero. For example, if f heads to 5 and g heads to -1, then f times g plus 3 heads to 5 times -1 plus 3, which is -2. Polynomials are the easy case: just substitute. Trouble starts when substitution makes the bottom zero. If the top stays nonzero, the quotient blows up and the limit fails. If both come out zero you meet 0 / 0, and that form never decides anything. It signals hidden common factors and tells you to simplify first.
Take (x squared - 9) / (x - 3) at x = 3. Substitution gives 0 / 0, so the quotient law stays silent. Factor the top as a difference of squares: (x - 3)(x + 3). Cancel the shared (x - 3), which is safe because x only nears 3 and never equals it. What remains is x + 3, and now substitution works: 3 + 3 = 6. Cubes surrender the same way: (x cubed - 1) / (x - 1) factors with (x - 1) on top and bottom, so the limit at 1 is 1 + 1 + 1 = 3.
Roots need a different escape called rationalising. Take (root(x + 9) - 3) / x at x = 0. Substitution gives 0 / 0 again. Multiply top and bottom by the conjugate root(x + 9) + 3, which changes the look but not the value. The top collapses to (x + 9) - 9, which is just x, while the bottom becomes x times (root(x + 9) + 3). Cancel the x, substitute 0, and you get 1 / (3 + 3), which is 1/6. The same move tames (root(x) - 2) / (x - 4) at 4: multiply by root(x) + 2, cancel (x - 4), and the limit is 1 / (2 + 2), which is 1/4.
The third escape is the squeeze. If g sits between two functions that head to the same number, g is forced there too. Since -x squared and x squared both head to 0, any g trapped between them near 0 also heads to 0. The wobble of x squared times sin(1 / x) dies the same way: sine never leaves its bounds, so the product sits between -x squared and x squared and must head to 0. Remember the order of work: substitute first, and only when you meet 0 / 0 do you factorise, rationalise or squeeze, then substitute again.
Zero over zero is a signal to simplify, and after factorising, rationalising or squeezing, substitution finishes the job.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.