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

• If m < n: limit = 0
• If m = n: limit = $\frac{leading coefficient of P}{(leading coefficient of Q)}$
• 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) $\frac{ax + b}{(sqrt(x^2 + ax + b)}$ + 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.