Figuring out when to use substitution or factorization for limits is an important part of understanding calculus! Let’s break it down!

1. When to Use Substitution:

• Direct Evaluation: If you can just plug in the number and get a real answer, use substitution! For example, to find $\lim_{x \to 3} (2x + 1)$, just put $3$ in the equation: $2(3) + 1 = 7$. Easy, right?
• Continuous Functions: If the function is smooth and doesn’t jump at the point you’re looking at, substitution works great!

2. When to Use Factorization:

• Indeterminate Forms: If plugging in a number gives you something confusing like $\frac{0}{0}$, it’s time to factor! For example, in $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$, both the top and bottom become zero. To make it simpler, factor the top to get $\frac{(x-2)(x+2)}{x-2}$ and then cancel out the $(x-2)$!
• Complex Expressions: If the limit has polynomials or fractions that make it hard to substitute directly, factorization can help make things clearer!

Learning to spot when to use these two methods can really help you find limits easily, and it’s super fun! Happy calculating! 🎉