Limits and Continuity
Core ideas
A limit describes the value a function approaches, not necessarily the value it attains. You must be fluent with one-sided limits, infinite limits, limits at infinity, and the existence criterion: the two-sided limit exists exactly when both one-sided limits agree. Computation tools include direct substitution, factoring, rationalizing, the squeeze theorem, and the special limits $\lim_{x\to0}\frac{\sin x}{x}=1$ and $\lim_{x\to0}\frac{1-\cos x}{x}=0$. L'Hôpital's rule applies only to indeterminate forms like $0/0$ or $\infty/\infty$ after you verify the form.
Continuity at a point requires three things: the function is defined there, the limit exists, and the two are equal. The Intermediate Value Theorem follows from continuity on a closed interval and guarantees a value is achieved between endpoint outputs - useful for proving a root exists.
Traps
- Cancelling a factor that is zero at the point changes the function's domain; the limit still exists but continuity may fail.
- A removable discontinuity (hole) versus a jump versus an asymptote: classify by the one-sided behavior.
- Do not apply L'Hôpital to a form that is not indeterminate; check first.
- Limits at infinity of rational functions depend on comparing degrees, not on plugging in large numbers.
Worked example
Evaluate $\lim_{x\to3}\frac{x^2-9}{x-3}$. Factor the numerator: $\frac{(x-3)(x+3)}{x-3}=x+3$ for $x\neq3$. The limit is $6$. The function has a removable discontinuity at $x=3$ because the simplified expression equals 6 there while the original is undefined.