Here's the easy way to write epsilondelta proofs for limits. Usually, the epsilondelta definition is taught in college calculus 1 (or real analysis in upperdivision pure math but not on AP calculus). I think it is the hardest topic in calculus 1. Let me know what you think!
Check out the Introduction to εδ definition of the limit    • epsilondelta definition ultimate int...  
More examples of writing εδ proofs    • limits with epsilondelta definition!...  

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0:00 proving limit of 1/2x+3 as x goes to 4
3:35 proving limit of sqrt(103x) as x goes to 2
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