EXAMPLE 2 Prove the following limit. lim 5x – 4 = 6 X → 2 SOLUTION 1. Preliminary analysis of the problem (guessing a value for 8). Let ɛ be a given positive number. We want to find a number 8 such that if 0 < |x – 2| < 8 then |(5x – 4) – 6| < e. But |(5x – 4) - 6| = |5x – 10| Therefore, we want 8 such that = 5 if 0 < ]x – 2| < 8 then 5 that is, if 0 < ]x – 2| < 8 then This suggests that we should choose 8 = ɛ/5. 2. Proof (showing that 8 works). Given ɛ > 0, choose 8 = ɛ/5. If 0 < < 8, then |(5x – 4) – 6| = < 58 = = E. Thus if 0 < |x – 2| < 8 then |(5x – 4) – 6| < e. Therefore, by the definition of a limit lim 5x – 4 = 6. X → 2 II

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.