Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: interview with Jim Simons.

Used Weierstrass's definition of a limit to prove that the limit, as x---> 1, of (x⁶-1)/(x-1) is equal to 6.

Then stated a proposition listing some properties of limits. I didn't bother deriving/proving this proposition from Weierstrass's definition. Instead of giving a proof, I illustrated how to apply the proposition to establish/prove certain limits.

For more details on the formal definition of a limit, read Section 2.4 in Stewart.