What is a Limit?

Published on Dec 20, 2019

๐Ÿ“Watch AP Calculus teacher Jamil Siddiqui, the Massachusetts Teacher of the Year 2019, explain graphical limits.

What is a limit?

A limit denotes theย behaviorย of a function as itย approachesย a certain value which is especially important in calculus.ย In mathematical terms, the limit is asking the question “What value is ‘y’ getting close to as ‘x’ approaches a number?” and its represented by the expression:

โ€‹โ€‹

Out loud, this would sound something like “the limit of f(x) as x approaches c”

  • The ‘lim’ shows that we’re finding the limit and not the value of the function
  • The ‘x โ†’c’ tells us the value that x is getting closer and closer to but never actually reaching
  • The ‘f(x)’ is meant to represent the functions you’re working with

Live Stream Replay: Algebraic Limits

Three Criteria for Limits to Exist

Study Guide: Determining Procedures for Determining Limits

1.) The limit of the function approaching from the right exists

  • Graphically, this would look like following a function to a certain point on the x-axis from the right

โ€‹โ€‹

2.) The limit of the function approaching from the left exists

  • Graphically, this would look like following a function to a certain point on the x-axis from the left

3.) The limit from the right equals the limit from the leftย โ€‹โ€‹

โ€‹โ€‹

Study Guide: Determining Limits using the Squeeze Theorem

When doesn’t a limit exist?

A limitย MUST meet all three criteria in order for it to exist, which is important to remember especially when FRQ’s come into play. If the limit from the left and the limit from the right are not equal, then the limit does not exist. An example of this would look something like this graph:

Here we can see that as we follow the function from the right hand side, it’s going towards y = 2. On the left hand side, we see that the function is approaching y = 1. Since 2 โ‰„ 1, the limit does not exist at x = 1.

โ€‹โ€‹

Study Guide: Exploring Types of Discontinuities

Limits Approaching Infinityย 

๐Ÿ‘‰ Watch AP Calculus teacher Suzanne Ferrell-Locke explain limits @ infinity.

โ€‹โ€‹

A limit approaching infinity is the same concept as a limit approaching any other number but the process is a little different.
โ€‹
This concept is relatively the same idea as end behavior and refers to the horizontal and oblique asymptotes that you might remember from Algebra 2 or Pre-Calc. There are three types of these types of limits you’re most likely going to see: top-heavy, middle-heavy, or bottom-heavy.

1.) Top-heavy

โ€‹This is when the degree of the polynomial on the top of the function is greater than the degree of the polynomial in the denominator In this case, the answer you are finding is all dependent on whether or not you’re being asked to be approaching negative or positive infinity. If you’re approaching positive infinity, then the limit of a top-heavy function will also be infinity and vice versa.

2.) Middle-heavy

This is when the degrees of the numerator and denominator are the same. To find the limit here, you’ll need to divide the leading coefficients of the numerator and denominator by each other.

โ€‹โ€‹

For example, here we see that the degrees of the top and bottom are 3 and that the leading coefficient of the numerator is 5 and 2 for the denominator.ย  Therefore, the limit of f(x) as x approaches positive infinity is 5/2.

3.) Bottom-heavy

This is when the degree of the denominator is greater than that of the numerator. If a function is bottom heavy, then the limit is zero, regardless if it is approaching positive or negative infinity

โ€‹Live Stream Replay: Continuity Part I

Live Stream Replay: Continuity Part II

๐Ÿ“ŒHere are some extra resources to help you out on limits approaching infinity!ย 

Live Stream Replay: L’Hospital’s Rule

โ™พ๏ธ 2020 calc ab+bc study guide

Prepare for the new 2020 AP Calculus AB + BC exam format with weekly lesson plans, study materials, and FRQ practice!

Best AP Calculus AB/BC Quizlet Decks By Unit

Best AP Calculus AB/BC Quizlet Decks By Unit

One of the most popular study sites used by AP students is Quizlet and for good reason! Quizlet combines the classic flashcard studying method with unique, fun games to learn vocabulary. However, the number of resources provided by Quizlet can make it challenging to...

AP Calculus AB/BC Free Response Questions (FRQ) โ€“ Past Prompts

AP Calculus AB/BC Free Response Questions (FRQ) โ€“ Past Prompts

Weโ€™ve compiled a sortable list of all the AP Calculus past prompts! The AP Calc FRQs are 60% of the exam including 2 long questions and 4 short questions. Itโ€™s important that you understand the rubrics and question styles going into the exam. Use this list to...

New Format Online AP Calculus Study Guide (AB+BC) – 2020 Changes

New Format Online AP Calculus Study Guide (AB+BC) – 2020 Changes

๐Ÿ‘‹๐Ÿผ Hi, I'm Jamil Siddiqui from East Bridgewater, MA! I'm an AP Calculus teacher and streamer at Fiveable. This year's exam is different than we expected, but I'm here to help. I've put together this study guide to help keep you on track while you are studying from...

0 Comments

Submit a Comment

Your email address will not be published. Required fields are marked *