Limits and Continuity Unit Review

When evaluating a limit:

• Read the direction. Is the input approaching a finite number, one side of a finite number, $+\infty$, or $-\infty$? • Substitute when meaningful.

• Real number: usually finished. • $0/0$: algebraic indeterminate form. • Nonzero over zero: one-sided sign analysis. • $\infty/\infty$: compare powers or divide by the dominant power. • $\infty-\infty$: combine or rationalize.

• Inspect the structure.

• polynomial factors $\to$ factor and cancel; • radicals $\to$ conjugate; • fractions inside fractions $\to$ common denominator; • absolute values/piecewise rules $\to$ one-sided cases; • bounded oscillation times zero $\to$ squeeze; • trig near zero $\to$ standard trig limits; • rational function at infinity $\to$ degrees/dominant terms; • radical at negative infinity $\to$ use $|x|$.

• State the conclusion precisely. Distinguish a finite number, $\mathrm{DNE}$, $+\infty$, and $-\infty$.