Limits & continuity
The idea
A limit records the value a function approaches, not the value it lands on. You already met the idea informally in precalculus; at university the limit becomes the load-bearing definition under everything else — derivatives, integrals, and series are all limits in disguise. Writing 'the limit as x → a of f(x) = L' claims that f(x) can be forced as close to L as you like by taking x close enough to a, with x = a itself deliberately ignored.
Continuity is the special case where the approach and the landing agree: f is continuous at a when f(a) exists, the limit exists, and the two are equal. The productive mental model is a checklist of three separate failures — hole, jump, or blow-up — each breaking a different clause. The classic misconception is that a 0/0 form means the limit does not exist. It means the opposite: the form is indeterminate, so you must do algebra (factor, rationalize, or use a known limit) before you can conclude anything.
A reliable workflow: substitute first, and if you get a number, you are done by continuity. If you get 0/0, manipulate the expression into an equivalent form where substitution works. If you get a nonzero number over zero, examine one-sided behavior, because the function is heading to plus or minus infinity.
Worked example
Let f(x) = (√(x + 4) − 2)/x for x ≠ 0. Find the limit of f(x) as x → 0, and state the value c you should assign to f(0) to make f continuous at 0.
- Substitute x = 0 first: the numerator is √4 − 2 = 0 and the denominator is 0, so the form is 0/0 — indeterminate, which licenses algebra rather than a conclusion.
- Rationalize by multiplying numerator and denominator by the conjugate √(x + 4) + 2. The numerator becomes (x + 4) − 4 = x, so f(x) = x divided by x(√(x + 4) + 2).
- Cancel the common factor x, valid because the limit process only cares about x near 0, never x = 0 itself. This leaves f(x) = 1/(√(x + 4) + 2) for all x ≠ 0.
- Now substitution is safe: as x → 0 the expression tends to 1/(√4 + 2) = 1/4.
- Sanity-check numerically: at x = 0.01, (√4.01 − 2)/0.01 ≈ 0.2498, which is hugging 0.25 as expected. Defining f(0) = 1/4 makes the function value equal the limit, which is exactly the definition of continuity at 0.
Answer. The limit is 1/4, so setting c = f(0) = 1/4 makes f continuous at 0.
Check your understanding
- Why does cancelling the factor x in a limit computation not contradict the rule against dividing by zero?
- What are the three distinct ways continuity can fail at a point, and what does each look like on a graph?
- How would you explain to a classmate why a 0/0 form is called indeterminate rather than undefined?
- Where in the definitions of the derivative and the definite integral is a limit secretly doing the work?
Build the foundations first
Limits & continuity builds on these concepts. If any feel shaky, start there.