Limits and Continuity
What you must understand
A limit describes the value a function approaches, not necessarily the value it takes. Master one-sided limits, the existence criterion (left limit equals right limit), and the algebra of limits. Know the standard techniques: direct substitution first, then factoring, rationalizing, or simplifying complex fractions. Recognize the special limits $\lim_{x\to0}\frac{\sin x}{x}=1$ and $\lim_{x\to0}\frac{1-\cos x}{x}=0$. Understand infinite limits and vertical asymptotes, and limits at infinity via leading-term comparison (degree rules for rational functions).
Continuity at a point requires three things: the function is defined there, the limit exists, and the two agree. Classify discontinuities as removable (hole), jump, or infinite. The Intermediate Value Theorem (IVT) guarantees a value is attained on an interval given continuity and an intermediate output; it is an existence tool, not a location tool.
Traps
- Assuming a limit equals the function value; holes are the classic case.
- Forgetting to check both one-sided limits at a piecewise breakpoint.
- Applying IVT without verifying continuity on the closed interval.
- Cancelling factors without noting the resulting domain restriction.
Worked example
Evaluate $\lim_{x\to3}\frac{x^2-9}{x-3}$. Factor: $\frac{(x-3)(x+3)}{x-3}=x+3$ for $x\neq3$. The limit is $6$, even though the function is undefined at $x=3$ (removable discontinuity).
Rule of thumb: substitution, then factor/rationalize, then special limits; always confirm continuity before invoking IVT.