When you start learning about limits in Pre-Calculus, you'll come across some common types of problems. Based on what I've learned, here’s a simple breakdown:

1. Finding Limits Algebraically

• These problems usually involve plugging in a number directly. For example, to find $\lim_{x \to 3} (2x + 1)$, you just put $3$ in for $x$. So, you get $2(3) + 1 = 7$. Simple, right?

2. Limits That Lead to Indeterminate Forms

• Sometimes, when you substitute, you might see something like $\frac{0}{0}$. That means you need some other tricks, like factoring. For example, in $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$, if we factor, we get $\frac{(x - 1)(x + 1)}{x - 1}$. When we cancel out the $(x - 1)$, we can then just look at $x + 1$ by substituting $1$, which gives us $2$.

3. One-Sided Limits

• These limits look at what happens when you approach a point from one side only. For example, $\lim_{x \to 2^-} (x^2 - 4)$ pays attention to values that are less than $2$ and will result in $0$ when computed.

4. Limits at Infinity

• This type looks at what happens as $x$ gets really big. For example, $\lim_{x \to \infty} \frac{2x^2 + 3}{5x^2 + 1}$ simplifies to $\frac{2}{5}$ as $x$ goes to infinity.

Practice Exercises

To help you learn, try solving these:

1. Find $\lim_{x \to 4} (x^2 - 16)$.
2. Find $\lim_{x \to 0} \frac{\sin x}{x}$.
3. Evaluate $\lim_{x \to -1} \frac{x^2 + 2x + 1}{x + 1}$.

As you practice, these limit problems will start to feel less confusing and more like a fun puzzle!