There’s a bit of a misconception in the graphing video!
At the 5:15 mark, Ben tells us that a vertical line on the speed vs time graph is impossible because it would be faster than the speed of light. Then goes on to show that there’s a particular slope that corresponds to the speed of light that we cannot exceed because nothing is faster than the speed of light.
Most of that statement is quite right! Nothing is faster than the speed of light, and the speed of light is indeed (slightly less than) 300 000 000 m/s.
However, as the graph is a speed vs time graph, the speed of light is actually a horizontal light that is very very high on the y-axis. A vertical line on the graph represents instant acceleration from one speed to another without “passing through” any other intermediate speeds. I believe this is also impossible, but it’s not related to the limit of the speed of light.