Cue Column: Key Questions

• What are the 7 indeterminate forms?
• How to resolve 1^infinity?
• What are the 3 conditions for continuity?
• How to classify discontinuities?

Notes Column: Limits are the foundation of calculus. The limit lim(x->a) f(x) = L means f(x) approaches L as x approaches a. For the limit to exist, the left-hand limit (LHL) must equal the right-hand limit (RHL).

The seven indeterminate forms are: 0/0, infinity/infinity, 0 * infinity, infinity - infinity, 0^0, 1^infinity, $infinity^0$. Each requires a specific resolution technique.

For 0/0: Try factorization, rationalization, standard limits, or L'Hopital's Rule. For infinity/infinity: Divide by highest power, or use L'Hopital's Rule. For 1^infinity: Use the formula e^(lim g(x)[f(x) - 1]) where f(x)->1 and g(x)->infinity. For 0 * infinity: Convert to 0/0 or infinity/infinity by rewriting. For infinity - infinity: Take common factors or rationalize.

Continuity at x = a requires: (1) f(a) is defined, (2) lim(x->a) f(x) exists, (3) lim(x->a) f(x) = f(a). Four types of discontinuity: removable, jump, infinite, oscillatory.

Summary: Master standard limits $\frac{sin x/x = 1, (e^x-1}{x}$ = 1, ln$\frac{1+x}{x}$ = 1), the 1^infinity formula, and the three continuity conditions. These cover 80% of JEE limit problems.