Limits
What a function approaches, even where it isn’t defined.
Pen and paper is fine · no calculator needed why?
Opens at level 31. You're level 1. You can read and practice here now.
the idea
A limit asks what value a function closes in on as x closes in on a number. The function doesn't need a value at that number: the limit is about the approach, not the arrival.
Three patterns cover this course. Getting 0/0 is not an answer; it signals a factor you can cancel. As x → ∞ (x growing without end), a fraction of polynomials follows its highest powers. Near 0, sin x and tan x are almost exactly x.
techniques
Factor, cancel, then put it in
- Put the number in. 0/0 means the top has the bottom as a factor.
- Factor the top and cancel the bottom’s factor.
- Put the number into what is left.
worked example
What is the limit of (x² − x − 6)/(x − 3) as x → 3?
- At x = 3 the top is 9 − 3 − 6 = 0, and so is the bottom.
- Factor: (x − 3)(x + 2). Cancel x − 3.
- What is left, x + 2, is 5 at x = 3.
Answer: 5
Keep only the highest powers
- Find the highest power of x on top and on the bottom, wherever it is written.
- Same power: divide the top one’s coefficient (the number in front) by the bottom one’s, in lowest terms.
- Higher power on the bottom: the bottom outgrows the top, so the limit is 0.
worked example
What is the limit of (6x² − 5)/(4x + 9x²) as x → ∞? Enter a fraction or a whole number.
- Both have x² as the highest power.
- Keep 6x² on top and 9x² on the bottom.
- 6/9 = 2/3.
Answer: 2/3
sin and tan near 0
- Near 0, sin u and tan u are almost exactly u. So replace sin(ax) or tan(ax) with ax.
- Cancel the x. The limit is a/b, in lowest terms.
worked example
What is the limit of sin(6x)/(4x) as x → 0? Enter a fraction or a whole number.
- For tiny x, sin(6x) ≈ 6x.
- So the limit is 6x/(4x), which is 6/4 = 3/2.
Answer: 3/2
watch out for
- Reading 0/0 as 0. It means the top and bottom share a factor: cancel it, then put the number in.
- Assuming anything with x on the bottom goes to 0. When the highest powers match, the limit is the ratio of their coefficients.
- Answering 1 for sin(5x)/x. The rule that sin u over u goes to 1 needs the same u on the bottom, so this limit is 5.
practice
Factor and cancel
worked example
What is the limit of (x² − 1)/(x − 1) as x → 1?
Answer: 2
- Putting in x = 1 gives 0/0, so factor the top: (x − 1)(x + 1).
- Cancel (x − 1) and put x = 1 into what is left, x + 1: 1 + 1 = 2.
Limits at infinity
worked example
What is the limit of (2x² + 6x − 1)/(−4x³ − 4x² + 1) as x → ∞? Enter a fraction or a whole number.
Answer: 0
- The bottom’s highest power (x³) is bigger than the top’s (x²), so the bottom grows much faster.
- The fraction shrinks toward 0, so the limit is 0.
sin x over x
worked example
What is the limit of sin(2x)/x as x → 0? Enter a fraction or a whole number.
Answer: 2
- For tiny x, sin(2x) ≈ 2x.
- So sin(2x)/x ≈ 2x/x = 2.