Definition of Limit

The definition of limit states that for a function f(x), the limit as x approaches a (written as lim x→a f(x)) is equal to L if and only if for every positive number ε, there exists a positive number δ such that if 0 < |x="" −="" a|="" />< δ,="" then="" |f(x)="" −="" l|="" />< />

Using the Definition of Limit to Show lim x→−1 (2x + 3) = 1

To show that lim x→−1 (2x + 3) = 1 using the definition of limit, we need to show that for every positive number ε, there exists a positive number δ such that if 0 < |x="" +="" 1|="" />< δ,="" then="" |2x="" +="" 5|="" />< ε="" −="" />

Proof

Let ε > 0 be given. We want to find a δ > 0 such that if 0 < |x="" +="" 1|="" />< δ,="" then="" |2x="" +="" 5="" −="" 1|="" />< />

We can start by manipulating the inequality |2x + 5 − 1| < />

|2x + 4| < />

2|x + 2| < />

|x + 2| < />

Now we can choose δ = ε/2. Then if 0 < |x="" +="" 1|="" />< δ,="" we="" />

|x + 2| < />

2|x + 2| < />

|2x + 4| < />

|2x + 5 − 1| < />

Thus, we have shown that for every positive ε, there exists a positive δ such that if 0 < |x="" +="" 1|="" />< δ,="" then="" |2x="" +="" 5="" −="" 1|="" />< ε.="" therefore,="" by="" the="" definition="" of="" limit,="" we="" can="" conclude="" that="" lim="" x→−1="" (2x="" +="" 3)="1." ε.="" therefore,="" by="" the="" definition="" of="" limit,="" we="" can="" conclude="" that="" lim="" x→−1="" (2x="" +="" 3)="" />