Techniques for Finding Limits: Substitution and Factorization

1. Substitution Method:

   • Let's say we want to find the limit as ( x ) gets close to 3 in the expression ( 2x + 1 ).
   • To do this, we just plug in ( x = 3 ).
   • So, we calculate ( 2(3) + 1 ), which equals 7.

2. Factorization Method:

   • Now, let’s find the limit as ( x ) approaches 2 for the fraction ( \frac{x^2 - 4}{x - 2} ).
   • First, we need to factor the top part (numerator): ( x^2 - 4 ) can be rewritten as ( (x - 2)(x + 2) ).
   • Next, we simplify the fraction. That gives us: [ \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \quad \text{(as long as ( x ) is not 2)} ]
   • Finally, we substitute ( x = 2 ) into ( x + 2 ).
   • This means ( 2 + 2 ), which is 4.