Now we can also find deltas algebraically for each epsilon we are faced with. This is way cool. Now try your hand at following this reasoning to prove that The actual evaluation of a limit at a particular value of x can involve direct substitution, algebraic manipulation followed by substitution, or other techniques and analysis. We are concerned only with its limit. This allows us to evaluate the limit.
Limits are the foundation of the calculus.
As previously stated, we cannot obtain a derivative without evaluating a limit. Now try your hand at evaluating some limits.