What a limit says
The limit of f(x) as x approaches a is the value that f(x) gets closer and closer to when x is near a but not equal to a. We write lim(x -> a) f(x) = L. The function does not even need to be defined at a. A limit studies the neighbourhood of a, never the point itself, so f(a) and the limit can be completely different, or f(a) can be missing.
Reading a limit from numbers
Take g(x) = (x^2 - 16)/(x - 4). At x = 4 it gives 0/0, which is undefined. Now try nearby inputs: g(3.9) = 7.9, g(3.99) = 7.99 and g(3.999) = 7.999 from the left, and g(4.001) = 8.001 from the right. Both sides head toward 8, so the limit is 8. A table suggests the answer; algebra proves it.
One-sided limits
The left-hand limit uses only x < a and the right-hand limit uses only x > a. The two-sided limit exists exactly when both one-sided limits exist and are equal. If the graph jumps, or if either side grows without bound, the two-sided limit does not exist. Never average two different one-sided values.
An algebra toolkit
- Try direct substitution first. A finite number means you are done.
- If you get 0/0, factor and cancel. Here x^2 - 16 = (x - 4)(x + 4), so for x not equal to 4 the fraction is just x + 4, and the limit is 8.
- If square roots cause the 0/0, multiply top and bottom by the conjugate.
- If you get a nonzero number over 0, expect the values to grow without bound, and check the sign on each side.
- Use the standard result that sin(x)/x approaches 1 as x approaches 0 (x in radians). By rescaling, sin(kx)/x approaches k.
Worked example
Find the limit as x approaches 9 of (sqrt(x) - 3)/(x - 9).
Substitution gives 0/0. Multiply top and bottom by the conjugate sqrt(x) + 3. The top becomes (sqrt(x))^2 - 9 = x - 9, which cancels the bottom, leaving 1/(sqrt(x) + 3) for x not equal to 9. As x approaches 9 this tends to 1/(3 + 3) = 1/6. A numeric check agrees: at x = 9.0001 the original fraction is about 0.1667.
Unbounded behaviour
As x approaches 0, the value of 1/x^2 grows without bound because the denominator is tiny and positive on both sides. We write this as +infinity to describe the behaviour, although strictly the limit does not exist as a real number.
Common mistakes
- Stopping at 0/0 and declaring the limit is 0 or does not exist. It is only a signal to do more algebra.
- Mixing up f(a) with the limit.
- Cancelling terms instead of factors, such as crossing out the x in (x + 4)/x.
- Forgetting that sin(x)/x -> 1 needs radians.