A limit in calculus is all about getting incredibly close to something without actually touching it. It describes the value a function approaches as its input gets closer and closer to a certain number.

Think about walking towards a wall. You can get very close: one foot away, one inch away, a millimeter away. You can keep cutting the distance in half forever. A limit is like asking, “If you could keep walking forever, what single point would you be aiming for?” The answer is the wall itself. In calculus, we're not interested in what happens at the wall (or the point), but what value the function is honing in on as it gets near.

Let's look at the function $f(x) = x + 2$. What is the limit of this function as $x$ approaches 3? We write this using the following notation:

It's easy to see that as $x$ gets closer to 3, $f(x)$ gets closer to 5. If $x$ is 2.9, $f(x)$ is 4.9. If $x$ is 3.01, $f(x)$ is 5.01. The value the function approaches is 5. So, the limit is 5.

This might seem obvious. Why not just plug in 3? Sometimes you can't. The real power of limits is for situations where a function is undefined at a specific point, but we still want to know how it behaves nearby.

Sometimes a function behaves differently depending on which direction you approach a point from. This is where one-sided limits come in.

• The left-hand limit is the value a function approaches as you come from the negative side (numbers smaller than the target). It's written with a minus sign: $\lim_{x \to c^-} f(x)$.
• The right-hand limit is the value a function approaches from the positive side (numbers larger than the target). It's written with a plus sign: $\lim_{x \to c^+} f(x)$.

For a general limit to exist, the left-hand limit must equal the right-hand limit. If they approach different values, the overall limit does not exist at that point.

In the graph above, as $x$ approaches 2 from the left, the function's value gets closer to 4. As $x$ approaches 2 from the right, the value is 1. Since these are different, the overall limit at $x=2$ doesn't exist.

Limits follow predictable rules that make them easier to work with. If the limits of $f(x)$ and $g(x)$ exist as $x \to c$, then:

1. Sum Rule: The limit of a sum is the sum of the limits. $\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$
2. Product Rule: The limit of a product is the product of the limits. $\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$
3. Quotient Rule: The limit of a quotient is the quotient of the limits, as long as the denominator's limit isn't zero. $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$

So how do we find limits? The first thing to try is always direct substitution. Just plug the number into the function. If you get a real number, that's your answer. If you get something like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, these are called indeterminate forms. They don't mean the limit doesn't exist, just that you need to do more work.

Techniques for indeterminate forms include:

• Factoring: Factor the numerator and denominator and cancel common terms.
• Rationalizing: Multiply by the conjugate to remove a square root.

For example, let's find $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$. Plugging in 2 gives $\frac{0}{0}$. So, we factor.

We can cancel the $(x-2)$ terms. Remember, we are only concerned with what happens near $x=2$, not at $x=2$, so $x-2$ is not zero. This leaves us with:

We can also ask what happens to a function as $x$ gets infinitely large, either positive ($\infty$) or negative ($-\infty$). This tells us about the function's long-term behavior, often called its end behavior.

For example, consider the function $f(x) = \frac{1}{x}$. What happens as $x$ gets huge?

As the denominator gets bigger and bigger (1/100, 1/1000, 1/1,000,000), the fraction gets smaller and smaller, approaching 0. So, the limit is 0.

When dealing with rational functions (a polynomial divided by another polynomial), a useful shortcut is to look at the highest power of $x$ in the numerator and denominator.

• If the highest power is in the denominator, the limit is 0.
• If the highest power is in the numerator, the limit is $\infty$ or $-\infty$.
• If the highest powers are the same, the limit is the ratio of their coefficients.

Now let's see what you've learned.