Limits

What it asks

Where is the function heading as x approaches a? Not what it equals there — the function need not even be defined at that point.

Try substituting first

Most of the time it just works. Only if you get 0/0 do you have work to do.

When you get 0/0

Factor and cancel, or multiply by the conjugate, then substitute again. 0/0 means “not enough information yet”, not zero.

One-sided

The limit exists only if both sides agree. Check both at a jump.