Introduction

Welcome to the FiveHive article for AP Calculus Unit 1.2: Defining Limits and Using Limit Notation!

Welcome back to our calculus adventure! 

Previously we talked about how we are going to use limits, but we never really said what they are. You might know the answer from a precalculus class, but in case you did not or need a refresher a limit is the value a function approaches at a certain $x$-value. 

This is only AP Calculus, so we will not be diving into the formal definition of a limit, but this definition will suffice for now and you will still be able to do amazing things with limits.

Limit Notation

If the limit as $x$ approaches $c$ of $f(x)=L$, it is written as 

$\underset{x\to c}{\lim }f(x)=L$.

Let us work on writing this down. You will need to master this notation.

Example 1:

Write the limit as $x$ approaches $0$ of $\frac{\sin x}{x}$ equals $1$.

$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$

Make sure there is a lim, under it the independent variable with a right pointing arrow to the value it approaches. Next to the lim, put the function and then put an equals sign with what the limit evaluates to on the right side.

Now that we know how to write limits, we can find them out in 3 ways: graphically, numerically, and algebraically/analytically.

Graphically:

In 1.3, we will learn how to interpret graphs to estimate limits. The graphs might be of some unfamiliar or complicated functions, but these graphs will contain information that will allow us to approach a value.

Numerically:

In 1.4, we will learn how to use tables to estimate limits. We probably won’t know the function, but we can make predictions.

Analytically:

In 1.5 & 1.6, we will use algebra skills to take limits of complicated functions at tricky values and make it a little bit easier for us to handle.

Before it is time to learn how to evaluate limits, just practice once more with limit notation. Be prepared, the difficulty will start to ramp up later.

Practice

Oh yeah! Now we’re really getting into the calculus. Can you feel it? Anyway, we have learned a lot in this topic, and now it is time to put your skills to the test.