How to Evaluate Limits

Method 1 — Direct substitution: Plug in x = a. If you get a number, that's the limit. Always try this first.

Method 2 — Factor and cancel: If you get 0/0, factor numerator and denominator, cancel common factors, then substitute.

lim(x→2) (x²-4)/(x-2) = lim(x→2) (x+2)(x-2)/(x-2) = lim(x→2) (x+2) = 4

Method 3 — Rationalize: If you see square roots, multiply by the conjugate.

Method 4 — L'Hôpital's Rule: If you get 0/0 or ∞/∞, take the derivative of top and bottom separately. lim f(x)/g(x) = lim f'(x)/g'(x).

Limits at infinity:

Divide by highest power of x. Leading term dominates.

lim(x→∞) (3x² + 2x)/(5x² - 1) = 3/5

Reference:

Wikipedia: Limit

https://en.wikipedia.org/wiki/Limit_(mathematics)