**Definition: **Calculus is often defined as “The Study of Limits”, so in order to understand calculus, it is important to first understand what a limit is. Let’s start with a really technical definition, and then talk about what it means. Say you have a function, , and an *x*-value, . We define “the limit of *f*(*x*) as *x* approaches *a*, written , to be *L* if:

For all , there exists a such that, if then

That’s certainly a mouthful! But what does it mean? Well, simply put, the limit of a function *f*(*x*) at a point *x*=*a* is simply what happens to the value of the function as *x* gets very, very close to a. Take a look at this image:

As you can see, as long as we pick an *x* that is very close to a, our function will be very close to *L*. If we want to get closer to *L*, we can just pick a smaller δ; the closer *x* gets to *a*, the closer *f*(*x*) gets to *L*.

So, how can we find a limit? There are a few methods you can try. The first is simply called *plugging in*, and all you do is plug the value of *a* into the functions. For many functions, .

**Example**: Find the limit as *x* approaches 5 of .

**Solution**:

Here, plugging in will work:

Another useful technique is *factoring*, which is useful for rational functions.

**Example**: Find the limit as *x* approaches 1 of .

**Solution**:

Let’s try plugging in again:

Unfortunately, it didn’t work! This function is undefined at *x*=1, so we can’t simply plug in to evaluate the limit. However, we can factor and cancel:

This technique works really well for functions that are fractions of polynomials. For fractions involving square roots, we’ll use a similar method called *multiplying by the conjugate*.

**Example**: Find the limit of as *x* approaches 9.

**Solution**: We can see very quickly that plugging in won’t work; we’ll get a 0 in the denominator again. But the numerator has this expression: , so we can try to simplify by multiplying by its conjugate.

Recall that the conjugate of a a radical expression switches the sign of the square root piece; here, the conjugate of is . When you multiply a radical expression by its conjugate, it will be rational; that is, the square root will go away. In order to not change the fraction, we have to multiply both the top and the bottom by the same number:

As you can see, multiplying by the conjugate allowed us to cancel the factor from the denominator that was giving us a 0, allowing us to plug into the function to obtain the result.

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