Pattern 1: 1^infinity Form (Appears almost every year) Questions like lim(x->0) [$\frac{sin x}{x}$]^(1/$x^2$) or lim(x->inf) [$\frac{x+a}{(x+b)}$]^(cx). Always apply the formula e^(lim g(x)*(f(x)-1)).

Pattern 2: Piecewise Function Continuity (Very frequent) Given f(x) defined in pieces, find parameter k for continuity at junction. Method: equate LHL = RHL = f(junction).

Pattern 3: Taylor-Based Limits (2-3 per year) Expressions like $\frac{sin x - x + x^3/6}{x}$^5 or $\frac{e^x - 1 - x - x^2/2}{x}$^3. Expand using Taylor to the needed order.

Pattern 4: Riemann Sum to Integral (Once per year) Recognize $\frac{1}{n}$*sum f$\frac{r}{n}$ = integral of f from 0 to 1. Common: $\frac{1}{n+r}$ terms, power sums.

Pattern 5: Greatest Integer Function Problems Find discontinuities of [f(x)], evaluate limits involving [x] or {x}. Requires careful one-sided analysis.

Pattern 6: L'Hopital with Parameter Finding If lim $\frac{sin ax + b*sin x}{x}$^3 is finite, find a and b. Method: Taylor expand, set coefficients of lower powers to zero.

Pattern 7: Continuity-Differentiability Connection Given f defined with |x| or x*sin$\frac{1}{x}$, determine continuity and differentiability at x = 0. Remember: continuous does not imply differentiable.

Scoring Strategy:

• Master the 1^infinity formula (guaranteed 1 question)
• Practice piecewise continuity (guaranteed 1 question)
• Learn Taylor expansions to $x^3$ term (handles most problems)
• Know GIF/fractional part properties (1 question every other year)