Calculating limits is an important idea in calculus. We can use algebra to make finding these limits easier. Let's break it down into simple parts.

What Are Limits?

A limit tells us the value a function gets closer to as we change the input (or the variable) to a certain number.

For example, when we say the limit of $f(x)$ as $x$ gets close to $a$ is $L$, we write it like this:

$$\lim_{x \to a} f(x) = L$$

Common Techniques

Here are some simple ways to find limits:

1. Direct Substitution: This means we directly put the value into the function. If it works (and we don’t get something weird like dividing by zero), we get our limit right away.

   For example, let’s find $\lim_{x \to 3} (2x + 1)$:

   • Substitute $x = 3$:
   $$2(3) + 1 = 6 + 1 = 7$$

   So, $\lim_{x \to 3} (2x + 1) = 7$.

2. Factoring: If direct substitution gives us a problem like $\frac{0}{0}$, we can try factoring.

   For example:

   $$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$

   If we factor it, it becomes:

   $$= \lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2}$$

   We can cancel out $(x - 2)$ (as long as $x \neq 2$):

   $$= \lim_{x \to 2} (x + 2) = 4$$

3. Rationalizing: When limits have square roots, we can multiply by the conjugate to make it easier.

   For example:

   $$\lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x}$$

   Here, we would multiply the top and bottom by the conjugate $\sqrt{x + 1} + 1$ to simplify.

Conclusion

These algebra techniques are great tools for figuring out limits. Getting good at these methods will help you understand calculus much better!

Practice using different functions so you can feel comfortable with each technique.