Polynomial/Rational Functions: For lim(x->infinity) P(x)/Q(x) where P has degree m and Q has degree n:

• If m < n: limit = 0
• If m = n: limit = (leading coefficient of P)/(leading coefficient of Q)
• If m > n: limit = +/- infinity

Method: Divide every term by x^(highest power in denominator).

Exponential vs Polynomial: Exponential growth dominates polynomial growth.

• lim(x->infinity) x^n / e^x = 0 for any fixed n
• lim(x->infinity) ln(x) / x^n = 0 for any n > 0

Growth Rate Hierarchy: (slowest to fastest) ln(x) << x^a (0 < a < 1) << x << x^2 << ... << x^n << e^x << x! << x^x

Useful trick for radicals: lim(x->infinity) (sqrt(x^2 + ax + b) - x): Multiply by conjugate. = lim(x->infinity) (ax + b)/(sqrt(x^2 + ax + b) + x) = a/2.

Warning: When x -> -infinity and you have sqrt(x^2), remember sqrt(x^2) = |x| = -x (since x is negative). This sign error is a very common JEE trap.