The formal definition of a limit is probably one of the most confusing things I have ever read. I read the definition about 5 times and even after the discussion in class I still don't really understand what it means. I prefer my own definition.
Real life example of a limit: Finding the acceleration of a moving object.
This is an example of a limit because as the object moves it is getting closer and closer to its final position, the same way a limit gets closer and closer to a specific number.
Problems:
1. Prove that the lim of 5x+2 = 22 as x approaches 4 by relating the gap.
Set f(x)-22 = |(5x+2)-22| = |5x-20| = 5*|x-4|, this means that the gap is 5 times as large as |x-4|.
2. Find a value for δ so that ε=0.4 for the function lim 10x+15 = 105 as x approaches 9.
First solve the limit by relating the gap: f(x) - 105 = |(10x+15)-105| = |10x-90| = 10|x-90|. Then set up the equation so that|x-90| < ε/10. Plug in 0.4, so 0.4/10 = 0.04. Now the gap has to be less than 0.04. So, |f(x)-105|<0.4 if 0<|x-90|<0.04
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