Limit laws
Limits of sums, differences, products and quotients behave as you would hope: take each limit, then combine. Quotients need a bottom limit that is not zero.
Last updated
Analogy
Like adding prices at a till: total the pieces, then combine.
Example
Limit of 3x^2 is 3 x 2^2 = 12.Limit of 1 is 1.Sum rule: 12 + 1 = 13.Polynomials: just substitute x = 2.
In practice
They let you evaluate most limits of polynomials by plain substitution.
Practice
Flashcards
- What is lim of 3x^2 + 1 as x tends to 2?13, by substituting x = 2.
- What does 0/0 tell you about a limit?It is indeterminate: simplify first and try again.
Interview questions
- Evaluate the limit of (x^2 - 9)/(x - 3) as x tends to 3 and explain each step.
- Substituting gives 0/0, which is indeterminate
- Factorise the top as (x - 3)(x + 3)
- Cancel the common factor for x not equal 3
- Substitute x = 3 into x + 3 to get 6
Questions people ask
Can I just substitute the value into the function?
For polynomials and other continuous functions, yes. If substituting gives a normal number, that is the limit. If you get 0/0, the rule does not decide the answer and you must simplify first, for example by factorising and cancelling common factors.
What can I do when substitution gives 0/0?
The form 0/0 is indeterminate, meaning the limit could be anything. Try factorising and cancelling, multiplying by a conjugate, or using L'Hopital's rule later. After cancelling, substitute again. For (x^2 - 9)/(x - 3) at 3, cancel to x + 3, giving 6.