Here is a video introduction to limits from Khan Academy.

If you watch through the whole video, he does some examples by making use of the graph, and also numerically using a calculator.

More details and examples are in these videos and practice problems:

Also from Khan Academy: limits using tables

and here are practice problems for limits using tables.

Also from Khan Academy: limits from graphs

There are several videos in that sequence, followed by practice problems.

Background:

There is a very nice discussion of the intuitive meaning of a limit, using examples “from real life”, at betterexaplained. You may want to read through this to see why we need limits and other uses of them that will appear later in calculus.