# High School Calculus/Evaluating Limits

### Evaluating Limits[edit]

What is a limit? A limit is a place on the graph that the function either does not touch or go past.

When evaluating a limit we may have to factor sometimes in order to get *L*. L is the point in which the function does not touch or go past.

Let's start off with a rather simple limit.

As you can see what we did was just plug 3 into the function to get *L*

This doesn't always work. This is easily shown in fractions.

I will show you two different ways to evaluate the limits. The first is by factoring and the second is by using L'Hopital's rule.

### Evaluating Limits by Factoring[edit]

This is a fairly simply concept, not something easily done. It is especially hard if you have a hard time identifying how polynomials can be rewritten.

**Ex.1**

This gives us

*L*

Let's look at how is factored

By factoring we now get

**Ex.2**

Factoring the polynomial we find that it equals

Let's use the factored in the limit equation.

As you can see the (x-2) will cancel each other out. Leaving us with

This type evaluating limits will take some time, but with practice can be done quickly.

### L'Hopital's Rule[edit]

This rule is my favorite way to solve limits with indeterminate form.

This way is a bit more advanced so I will cover it briefly, but I will show some examples and the idea behind it. This is probably something you will learn in Calculus II

When you have a limit that you have confirmed that is in indeterminate form you can use L'Hopital's Rule.

This is the rule

When , , , , , or use L'Hopital's rule. Which is

**Ex. 1**

Now that we have identified that it is in an indeterminate form we use L'Hopital's rule

This is an extremely simplified form of how this rule is used. It is a really nice way to solve limit problems that give you indeterminate forms.